Linear Inequalities

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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 − 4x ≤ -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 − 4x − 7 ≤ -13 − 7

         -4x ≤ -20

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

 -4x ÷ -4 ≥ -20 ÷ -4