Multiple Choice
Solve and graph the following absolute value inequalities. Express the solution in interval notation.
(A)
17
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9. Inequalities and Absolute Value
Absolute Value Inequalities
Multiple Choice
Solve and graph the following absolute value inequalities. Express the solution in interval notation.
(A)
A
B
OR
C
OR
D
Verified step by step guidance1
Understand that the inequality \(\left|2x + 0.6\right| > 2.6\) means the expression inside the absolute value, \$2x + 0.6$, is either greater than 2.6 or less than -2.6. This is because the absolute value measures distance from zero, so being greater than 2.6 means being outside the interval between -2.6 and 2.6.
Set up two separate inequalities to solve:
1) \$2x + 0.6 > 2.6$
2) \$2x + 0.6 < -2.6$
Solve the first inequality \$2x + 0.6 > 2.6\( by isolating \)x\(:
- Subtract 0.6 from both sides: \)2x > 2.6 - 0.6$
- Simplify the right side: \$2x > 2.0$
- Divide both sides by 2: \(x > 1.0\)
Solve the second inequality \$2x + 0.6 < -2.6\( by isolating \)x\(:
- Subtract 0.6 from both sides: \)2x < -2.6 - 0.6$
- Simplify the right side: \$2x < -3.2$
- Divide both sides by 2: \(x < -1.6\)
Combine the two solution sets to express the solution in interval notation: \(x < -1.6\) or \(x > 1.0\). This corresponds to the union of intervals \((-\infty, -1.6)\) and \((1.0, \infty)\). When graphing, shade the regions to the left of -1.6 and to the right of 1.0 on the number line.
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