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The rules for solving inequalities are similar to those for solving linear equations. However, there is one exception when multiplying or dividing by a negative number.

To solve an inequality, we can:

• Add the same number to both sides.

• Subtract the same number from both sides.

• Multiply both sides by the same positive number.

• Divide both sides by the same positive number.

**• Multiply** both sides by the same **negative number** and **reverse the sign****.**

**• Divide** both sides by the same **negative number** and **reverse the sign****.**

* Example: *

Solve *x* + 7 < 15

* Solution: *

* x* + 7 < 15

*x + * 7 – 7 < 15 – 7

*x* < 8

* Example* :

Solve *x* – 6 > 14

* Solution: *

* x* – 6 > 14

*x * – 6*+* 6 > 14 + 6

*x* > 20

** Example : **

Solve the inequality *x* – 3 + 2 < 10

* Solution: *

* x* – 3 + 2 < 10

*x* – 1 < 10

*x* – 1 + 1 < 10 + 1

*x* < 11

* Example*

Solve the inequality 7 – *x* < 9

* Solution: *

7 – *x* < 9

7 – *x* – 7 < 9 – 7

– *x* < 2

*x* > –2** (remember to reverse the symbol when multiplying by –1 ) **

* Example*

Solve the inequality 12 > 18 – *y*

* Solution: *

12 > 18 – *y
*18 –

18 –

–

* Example: *

Solve > 3

* Solution: *

> 3

× 5 > 3 × 5

* x* > 15

* Example: *

Solve

* Solution: *

If an equation has like terms, we simplify the equation and then solve it. We do the same when solving inequalities with like terms.

* Example*

Evaluate 3*x* – 8 + 2*x* < 12

* Solution: *

3*x* – 8 + 2*x* < 12

3*x *+ 2*x* < 12 + 8

5*x* < 20

*x* < 4

** Example: **

Evaluate 6*x* – 8 > *x* + 7

* Solution: *

6*x* – 8 > *x* + 7

6*x – x* > 7 + 8

5*x* > 15

*x* > 3

* Example: *

Evaluate 2(8 – *p*) ≤ 3(*p* + 7)

* Solution: *

2(8 – *p*) ≤ 3(*p* + 7)

16 – 2*p* ≤ 3*p* + 21

16* – *21 ≤ 3*p* + 2*p
*

Solving Linear Inequalities -

Examples are shown of how to solve linear inequalities

Students learn that when solving an inequality, such as --3x is less than 12, the goal is the same as when solving an equation: to get the variable by itself on one side.

Note that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be switched.

For example, to solve --3x is less than 12, divide both sides by --3, to get x is greater than -4.

And when graphing an inequality on a number line, less than or greater than means an open dot, and less than or equal to or greater than or equal to means a closed dot.

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