Linear Inequalities

**Linear Inequalities**

Solving a linear inequality is similar to solving a linear equation. The only difference is that when you multiply or divide both sides by a negative number, you must change the direction of the inequality.

6 > 4

-1(6) < -1(4)

-6 < -4

To graph an inequality, mark its position and direction on a number line.

*x* < 3

The open circle indicates that the value of 3 is not part of the inequality.

*x* ≤ 3

The closed circle now indicates that the value of 3 is part of the inequality.

To express an inequality in interval notation:

- Include the span of numbers included in the group from left to right, separated by a comma.
- Use parenthesis next to a number that is excluded from the group. Use a bracket when the number is included in the group.
- Use the infinity or negative infinity symbols to denote when the group goes on indefinitely. Always use round parentheses symbols with infinity.

*x* < 3

(-∞, 3)

*x* ≤ 3

(-∞, 3]

**Example **

7 − 4*x* ≤ -13

As you would with an equation, add or subtract terms to get all variables on one side and all constants on the other.

7 − 4*x* − 7 ≤ -13 − 7

-4*x* ≤ -20

Now divide both sides by the coefficient of *x*. (Careful, the coefficient in this example is negative!)

-4*x* ÷ -4 ≥ -20 ÷ -4

**Practice**

*x*+ 3 > 19

*x*+ 2 < 5

*x*− 6

*x*+ 6 ≥ 9

*x*− 4

*x*> 20

*x*− 1) ≥ 5(

*x*+ 3) + 2

**Answers**