Page 619 - Algebra 1

Some absolute-value inequalities have variable expressions inside the absolute-value symbols. The expression inside the absolute-value symbols can be positive or negative.
The inequality x + 1 < 3 represents all numbers whose distance from -1 is less than 3.
3 units 3 units
-4 -2 0 2
The inequality x + 1 > 3 represents all numbers whose distance from -1 is greater than 3.
3 units 3 units
-4 -2 0 2
Rules for Solving Absolute-Value Inequalities
For an inequality in the form K < a, where K represents a variable expression and a > 0, solve -a < K < a or K > -a AND K < a.
For an inequality in the form K > a, where K represents a variable expression and a > 0, solve K < -a OR K > a.
Similar rules are true for K ≤ a or K ≥ a.
Example
3
Solving Inequalities with Operations Inside Absolute-Value Symbols
Solve each inequality. Then graph the solution.
a. x-5 ≤3
SOLUTION
Use the rules for solving absolute-value inequalities to write a compound inequality.
x - 5 ≤ 3
x - 5 ≥ -3 AND x - 5 ≤ 3 Write the compound inequality.
+__5 +__5 +__5 +__5 Addition Property of Inequality x≥2 AND x≤8 Simplify.
Now graph the inequality. b. x+7 >3
SOLUTION
0 2 4 6 8
Use the rules for solving absolute-value inequalities to write a compound inequality.
x + 7 > 3
x+7<-3 OR x+7>3
-__7 -__7 -__7 -__7 x < -10 OR x > -4
Now graph the inequality.
-12 -8 -4
Writethecompoundinequality. Subtraction Property of Inequality Simplify.
0 4 8
604 Saxon Algebra 1