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Working with Linear Inequalities
(single variable)
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Click Basic Inequalities for introductory information.

statement
If you can solve a linear equation, you can solve a linear inequality. The process is the same, with one important exception ...

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... when you multiply (or divide) an inequality by a negative value,
you must change the direction of the inequality.

Let's see why this "exception" is actually needed.

We know that 3 is less than 7.
Now, lets multiply both sides by -1.
Examine the results (the products).
... written 3 < 7.
... written (-1)(3) ? (-1)(7)
... written -3
? -7

On a number line, -3 is to the right of -7, making -3 greater than -7.
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-3 > -7
We have to reverse the direction of the inequality, when we multiply by a negative value, in order to maintain a "true" statement.

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ex1

Solve and graph the solution set of:   4x < 24
Proceed as you would when solving a linear equation:
Divide both sides by 4.

Note: The direction of the inequality stays the same since we did NOT divide by a negative value.

Graph using an open circle for 6 (since x can not equal 6) and an arrow to the left (since our symbol is less than).

iqmath7

circleg3

CHECK: pick 1: 1 < 6 TRUE
pick 7: 7 < 6 FALSE



ex2

Solve and graph the solution set of:   -5x greaterequala 25
Divide both sides by -5.

Note: The direction of the inequality was reversed since we divided by a negative value.

Graph using a closed circle for -5 (since x can equal -5) and an arrow to the left (since our final symbol is less than or equal to).

iqmath8

circlegraph8a

CHECK: pick -7: -7 < -5 TRUE
pick 0: 0 < -5 FALSE



ex3

Solve and graph the solution set of:   3x + 4 > 13
Proceed as you would when solving a linear equation:
Subtract 4 from both sides.
Divide both sides by 3.

Note: The direction of the inequality stays the same since we did NOT divide by a negative value.

Graph using an open circle for 3 (since x can not equal 3) and an arrow to the right (since our symbol is greater than).

iqmath1

circleg3

CHECK: pick 4: 4 > 3 TRUE
pick 0: 0 > 3 FALSE



ex7

Solve and graph the solution set of:   9 - 2x lessequal 3
Subtract 9 from both sides.
Divide both sides by -2.

Note: The direction of the inequality was reversed since we divided by a negative value.

Graph using a closed circle for 3 (since x can equal 3) and an arrow to the right (since our symbol is greater than or equal to).

iqmath4

circleg4

CHECK: pick 4: 4 > 3 TRUE
pick 0: 0 > 3 FALSE

ex5

Solve and graph the solution set of:   5 - 2x lessequal 13 + 2x
Add 2x to both sides.
Subtract 13 from both sides.
Divide both sides by 4.

Note: There was no multiplication or division by a negative value, so the inequality symbol did not get reversed.

Graph using a closed circle for -2 (since x can equal -2).

CHECK: pick 0: 0 > -2 TRUE
pick -4: -4 > -2 FALSE

iqpuc8

iq8graph



ex6

Solve and graph the solution set of:   4x + 10 < 3(2x + 4)
Distribute across parentheses.
Subtract 4x from both sides.
Add -12 to both sides.
Divide both sides by 2.

Note: There was no multiplication or division by a negative value, so the inequality symbol did not get reversed.

Graph using a closed circle for -2 (since x can equal -2).

CHECK: pick 0: 0 > -1 TRUE
pick -3: -3 > -1 FALSE

iq9prob

iq9graph


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hint gal
Yes, there is a way to determine solutions for inequalities on your graphing calculator. Click the calculator at the right to see how to use the calculator with single variable inequalities.
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For calculator help with inequalities
(single variable)

click here.



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