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www.mathcentre.ac.uk/.../Solving%20Inequalities.mp4

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    The expression 5X
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    minus 4. Greater than
    two X plus 3 looks like an
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    equation, but with the equal
    sign replaced by an Arrowhead.
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    This denotes that the.
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    Part on the left, 5X minus four
    is greater than the part on the
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    right 2X plus 3.
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    We use four symbols to denote in
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    Equalities. This symbol
    means is greater than.
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    This symbol means is greater
    than or equal to.
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    This symbol means is less than.
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    On this symbol means is less
    than or equal to.
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    Notice that the Arrowhead
    always points to the
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    smaller expression.
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    In Equalities can be
    manipulated like equations
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    and follow very similar
    rules.
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    But there is one
    important exception.
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    If you add the same number to
    both sides of an inequality, the
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    inequality remains true. If you
    subtract the same number from
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    both sides of the inequality, it
    remains true. If you multiply or
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    divide both sides of an
    inequality by the same positive
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    number, it remains true.
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    But if you multiply or divide
    both sides of an inequality by a
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    negative number. It's no longer
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    true. In fact, the inequality
    becomes reversed. This is quite
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    easy to see because we can write
    that four is greater than two.
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    But if we multiply both sides of
    this inequality by minus one, we
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    get minus 4.
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    Is less than minus 2?
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    We have to reverse the
    inequality.
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    This leads to difficulties
    when dealing with variables
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    because of variable can
    be either positive or
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    negative. Look at these two
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    inequalities. X is greater than
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    one. And X squared.
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    Is greater than X.
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    Now clearly if X squared is
    greater than ex, ex can't be 0.
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    So it looks as if we ought to be
    able to divide both sides of
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    this inequality by X. Giving us.
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    X greater than one, which is
    what we've got on the left.
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    But in fact we can't do this.
    These two inequalities are not
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    the same. This is because X
    can be negative.
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    Here we're saying that X is
    greater than one, so X must be
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    positive. But here we have to
    take into account the
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    possibility that X is negative.
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    In fact, the complete solution
    for this is X is greater than
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    one or X less than 0.
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    Because obviously if X is
    negative, then X squared is
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    always going to be greater than
    X. I'll show you exactly how to
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    get the solution for this type
    of inequality later on.
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    Great care really has to be
    taken when solving inequalities
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    to make sure that you don't
    multiply or divide by a negative
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    number by accident. For example,
    saying that X is greater than Y.
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    Implies. That X
    squared is greater than Y
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    squared only if X&Y are
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    positive. I'll start
    with a very simple
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    inequality. X +3 is
    greater than two.
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    To solve this, we simply need
    to subtract 3 from both sides.
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    If we subtract 3 from the left
    hand side were left with X. If
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    we subtract 3 from the right
    hand side were left with minus
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    one and that is the solution
    to the inequality.
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    In Equalities can be represented
    on the number line.
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    Here are solution is X is
    greater than minus one.
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    So we start at minus one.
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    And this line shows the range of
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    values. The decks can take.
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    I'm going to put an open circle
    there. That open circle denotes
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    that although the line goes to
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    minus one. X cannot actually
    equal minus. 1X has to be
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    greater than minus one.
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    Let's have a
    look at another
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    one.
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    4X plus 6.
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    Is greater than 3X plus 7.
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    First of all, I'm going to
    subtract 6 from both sides, so
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    we get 4X on the left, greater
    than 3X plus one.
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    And now I'm going to subtract 3
    X from both sides, which gives
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    me X greater than one.
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    And again, I can represent this
    on the number line.
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    X has to be greater than one.
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    But X cannot equal 1.
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    Another example is 3X minus
    five is less than or
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    equal to 3 minus X.
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    This time I need to add 5 to
    both sides which gives me 3X is
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    less than or equal to.
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    8 minus X.
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    And then I need to add extra
    both sides, which gives me 4X
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    less than or equal to 8.
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    Finally, I can divide both sides
    by two, which gives me X is less
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    than or equal to two.
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    And on the number line.
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    X is less than or equal to two,
    so we go this way.
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    And this time I'm going to do a
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    closed circle. This denotes that
    X can be equal to two.
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    Now I'd like to look at
    the inequality minus 2X is
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    greater than 4.
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    In order to solve this
    inequality, we're going to have
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    to divide both sides by minus 2.
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    So we get minus two X divided by
    minus two is X.
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    I've got to remember because I'm
    dividing by a negative number to
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    reverse the inequality.
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    And four divided by minus
    two is minus 2, so I get
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    X is less than minus 2.
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    There's often more than one way
    to solve an inequality.
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    And I can just solve this
    one again by using a
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    different method, so we have
    -2 X is greater than 4.
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    If we add 2X to both sides we
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    get. Zero is greater than
    4 + 2 X.
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    And then if we subtract 4 from
    both sides, we get minus four is
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    greater than two X.
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    And we can divide through by two
    again getting minus two is
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    greater than X.
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    And saying that X is less than
    minus two is the same thing as
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    saying minus two is greater than
    X, so we've solved this
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    inequality by do different
    methods. The second one avoids
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    dividing by a negative number.
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    In Equalities often appear in
    conjunction with the modulus
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    symbol. For instance.
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    We say MoD X is less than two.
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    The modular symbol denotes that
    we have to take the absolute
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    value of X regardless of sign.
    This is just the magnitude of X.
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    And it is always
    positive. So for
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    instance, MoD 2 is
    equal to 2.
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    And MoD minus two is
    also equal to two.
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    If the absolute value of X is
    less than two, then X must lie
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    between 2:00 and minus two. We
    write minus two is less than X,
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    is less than two.
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    We can show this on the
    number line.
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    X has to lie between minus two
    and two, but it can't be too
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    itself. This shows the range
    of values that ex can take.
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    If MoD X is greater than or
    equal to five, we have the
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    absolute value of X must be
    greater than or equal to five,
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    which means that X is going to
    itself is going to be greater
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    than or equal to five or less
    than or equal to minus five. We
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    write X less than or equal to
    minus five or X greater than or
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    equal to 5.
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    And on the number line.
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    X can take the value 5, so we do
    a closed circle.
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    And it can take the
    value minus 5.
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    Now I want to look at
    another slightly more
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    complicated modulus one.
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    We have MoD X minus 4.
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    Less than three.
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    The modulus sign shows that
    the absolute value of X minus
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    four is less than three. This
    means that X minus four must
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    lie between minus three and
    three, so we write minus
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    three less than X minus four
    less than three.
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    This is what we call a double
    inequality of women's treated as
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    two separate inequalities. So on
    the left we have minus three is
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    less than X minus 4.
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    By adding four to both sides, we
    get one is less than X. On the
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    right we have X minus four is
    less than three.
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    And again we had four to both
    sides to get. X is less than 7.
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    So the solution to this
    particular inequality is X is
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    greater than One X is less
    than Seven. We write 1 less
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    than X less than Seven, and
    again I'll show you that on
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    the number line.
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    X lies between one and Seven,
    but it can't be either.
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    Now let's solve
    MoD. 5X. Minus 8
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    is less than or
    equal to 12.
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    We're saying here that the
    absolute value of 5X minus 8 is
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    less than or equal to 12.
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    So 5X minus 8.
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    Must be less than 12.
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    Or greater than minus 12.
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    We write minus 12 is less than
    or equal to 5X minus 8.
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    Is less than or equal to 12?
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    Again, we have a double
    inequality on the left, we have
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    minus 12 is less than or equal
    to 5X minus 8.
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    We add it to both sides, which
    gives us minus four is less than
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    or equal to 5X.
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    And then we divide both
    sides by 5, which gives
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    us minus four fifths is
    less than or equal to X.
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    On the right we have the
    inequality 5X minus 8 is less
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    than or equal to 12.
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    So we write 5X minus 8 less than
    or equal to 12.
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    We had eight to both sides,
    which gives us 5X is less than
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    or equal to 20.
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    And we divide both sides
    by 5, which gives us X is
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    less than or equal to 4.
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    So our final answer is minus 4
    over 5 is less than or equal to
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    X. Which in turn is less
    than or equal to 4.
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    And we can show this
    on the number line.
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    Minus four fifths is about here.
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    Let me go through to four.
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    And because it's less than or
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    equal to. We use
    a closed circle.
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    In Equalities can be solved
    very easily using graphs,
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    and if you're in any way
    unsure about the algebra it
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    can could be a good idea to
    do a graph to check. Let me
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    show you how this works.
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    We take the inequality 2X, plus
    three is less than 0.
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    Now this inequality can be
    solved very easily doing
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    algebra, but it makes a good
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    example. The first thing that we
    need to do is to draw the graph
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    of Y equals 2X plus 3.
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    And I've got this graph here.
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    Note that it's the equation of
    a straight line.
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    It has a slope of two
    and then intercept on
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    the Y axis of three.
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    On the X axis.
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    Why is equal to 0 so that
    where the line cuts the X
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    axis Y is equal to 0?
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    Above the X axis Y is greater
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    than 0. And below the X axis Y
    is less than 0.
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    So when we say that we want 2X
    plus three less than 0.
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    On this graph, that means why is
    less than zero, so we're looking
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    for the points where the line is
    below the X axis.
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    In other words, where X is less
    than minus one and a half, and
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    this is the solution to the
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    inequality. And we can mark
    this on the graph using the
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    X axis as the number line.
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    This technique can also be
    used with modulus inequalities
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    and here using a graph
    can be very helpful.
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    Take for example the inequality.
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    MoD X minus two is less than 0.
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    Again, we need to plot the graph
    of Y equals MoD X minus 2.
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    This is the graph of Y equals
    MoD X minus 2.
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    For those of you who are not
    familiar with modulus functions,
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    it might look a little bit
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    strange. On the right we have
    part of the graph of Y equals X
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    minus 2. And on the left,
    where X is less than zero, we
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    have part of the graph of Y
    equals minus X minus two.
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    This is because the modulus
    function changes the sign of
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    X when X is negative.
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    Again, we're looking for MoD X.
    Minus two is less than 0.
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    So we want the places where Y is
    less than zero, which is between
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    X equals minus two and X equals
    +2, and again this is the
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    solution to our problem.
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    So we say minus two less than
    X less than two.
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    Again, we can mark this on the
    graph using the X axis as the
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    number line. Quadratic
    inequalities need
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    handling with care.
    Let's solve X
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    squared minus three
    X +2 is
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    greater than 0.
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    Note that all the terms are on
    the left hand side.
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    And on the right hand side we
    just had zero, exactly as with
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    the quadratic equation before
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    you solve it. This expression
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    factorizes too. X minus
    two X minus one.
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    Now this is a quadratic
    equation. We would simply say
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    right X equals 2 or X equals 1
    and that's it.
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    But we've got a bit more work to
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    do here. Weather this expression
    is greater than zero is going to
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    depend on the sign of each of
    these two factors. We sort this
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    out by using a grid.
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    The points
    that were
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    checks equals.
    X minus 2 equals 0 and X minus
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    one equals 0 and marked in, so
    this is one and two.
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    We put the two factors on the
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    left. And their product.
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    Now.
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    When X is less than one, both X
    minus one and X minus two are
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    going to be negative.
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    So when you multiply them
    together, their product is going
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    to be positive.
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    When X is greater than one but
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    less than two. X minus one is
    going to be positive.
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    But X minus two is going to be
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    negative. So when you multiply
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    them together. The product will
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    be negative. Finally, when X is
    greater than two, both X minus
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    one and X minus two will be
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    positive. And if you multiply
    them together, their product
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    will also be positive.
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    We are looking for.
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    X minus two times X minus one to
    be greater than 0.
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    This occurs when it's positive.
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    And our grid shows that this
    happens when X is less than one.
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    Or when X is greater than two?
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    So we write in our answer.
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    Which is X is less than one
    or X is greater than two.
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    And on the number line.
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    X must be less than one.
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    So I put a circle to show
    that it can't be 1.
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    And X can also be greater
    than two.
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    Here's another
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    quadratic. Minus two
    X squared plus 5X
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    plus 12 is greater
    than or equal to 0.
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    I don't like having a negative
    coefficient of X squared, so I'm
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    going to multiply this whole
    thing through by minus one,
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    remembering to change the
    direction of the inequality as I
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    do. This gives us.
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    Two X squared minus 5X minus 12
    is less than or equal to 0.
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    This expression factorizes to 2X
    plus three times X minus four,
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    so that is less than or equal
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    to 0. Again, I'm going to
    do a grid.
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    This factor is zero
    when X is minus
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    three over 2.
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    This fact is zero when X is 4.
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    We write in the two factors.
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    And we right in the product.
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    When X is less than minus three
    over 2, both 2X plus three and
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    X minus four and negative.
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    So their product is positive.
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    When X lies between minus three
    over two and four.
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    2X plus three is positive.
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    But X minus four is still
    negative, so their product
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    is negative.
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    When X is greater than four,
    both 2X plus three and X minus
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    four are positive.
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    So their product is positive.
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    We are looking for 2X plus three
    times X minus four to be less
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    than or equal to 0.
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    In other words, this expression
    has to be either 0 or negative.
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    This occurs.
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    When X lies between minus three
    over two and four, and it can
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    equal either number. So we have
    minus three over 2 is less than
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    or equal to X is less than or
    equal to 4.
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    And on the number line.
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    Minus three over 2 is here.
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    Four is here.
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    And I've done filled
    circles because we have
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    less than or equal to.
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    Quadratic inequalities can
    also be solved graphically.
  • 23:23 - 23:31
    Let's solve X squared minus
    three X +2 is greater
  • 23:31 - 23:32
    than 0.
  • 23:34 - 23:39
    As with the linear equalities
    inequalities, we have to plot
  • 23:39 - 23:44
    the graph of Y equals X squared
    minus three X +2.
  • 23:45 - 23:52
    This factorizes to give Y equals
    X minus one times X minus 2.
  • 23:53 - 23:55
    The graph looks like this.
  • 23:56 - 24:01
    Because it's a quadratic, it's a
    parabola. Are U shaped curve?
  • 24:02 - 24:04
    And it crosses the X axis where
  • 24:04 - 24:09
    X equals 1. Because of the
    factor X minus one and where
  • 24:09 - 24:12
    X equals 2 because of the
    factor X minus 2.
  • 24:13 - 24:19
    Now we're looking for X squared
    minus three X +2 to be greater
  • 24:19 - 24:24
    than 0. This is where Y
    is greater than zero. In
  • 24:24 - 24:27
    other words, the part of
    the graph that is above
  • 24:27 - 24:31
    the X axis, which are the
    two arms of the you here.
  • 24:33 - 24:36
    This occurs where X is less than
  • 24:36 - 24:41
    one. And where X is greater
    than two, so we can write
  • 24:41 - 24:43
    that in as our solution.
  • 24:46 - 24:52
    And we can mark this
    in using the X axis
  • 24:52 - 24:54
    as the number line.
  • 24:56 - 25:00
    I'll
    do
  • 25:00 - 25:05
    one
    more
  • 25:05 - 25:07
    quadratic
  • 25:07 - 25:09
    inequality.
  • 25:10 - 25:14
    X squared Minus X
  • 25:14 - 25:18
    minus 6. So less than or
    equal to 0.
  • 25:23 - 25:27
    Again, we need to plot
    the graph of Y equals X
  • 25:27 - 25:29
    squared minus X minus 6.
  • 25:30 - 25:32
    The expression factorizes.
  • 25:33 - 25:36
    To X minus three.
  • 25:36 - 25:40
    X +2 And the graph
  • 25:40 - 25:47
    looks like this. Similar
    to the previous
  • 25:47 - 25:48
    graph.
  • 25:49 - 25:55
    We have The factor X +2 the line
    crosses the point at X equals
  • 25:55 - 25:59
    minus two and for the factor X
    minus three, the curve crosses
  • 25:59 - 26:01
    the point at X equals 3.
  • 26:02 - 26:06
    And we're looking for where X
    squared minus X minus six is
  • 26:06 - 26:08
    less than or equal to 0.
  • 26:09 - 26:14
    In other words, why must lie on
    the X axis or below it?
  • 26:15 - 26:20
    This part of the curve and that
    occurs between the points of X
  • 26:20 - 26:25
    equals minus two and X equals 3.
    So we can say that minus two is
  • 26:25 - 26:30
    less than or equal to X, which
    is less than or equal to 3.
  • 26:31 - 26:37
    And we can put this in again
    using the X axis is the
  • 26:37 - 26:41
    number line from minus 2
    using a closed circle because
  • 26:41 - 26:44
    2 - 2 is included to +3.
Title:
www.mathcentre.ac.uk/.../Solving%20Inequalities.mp4
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