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Chapter 0: Problem 78

Solve each absolute value inequality. $$5|2 x+1|-3 \geq 9$$

Short Answer

Expert verified
The solution to the inequality is \(x \geq 0.7\) or \(x \leq -1.7\).

Step by step solution

01

Isolate the Absolute Value

Firstly isolate the absolute value term on one side of the inequality. You're starting with the inequality \(5|2x + 1| - 3 \geq 9\). By adding 3 to both sides, all terms without absolute value are shifted to the right side: \(5|2x + 1| \geq 12\). Then, divide both sides by 5 to fully isolate the absolute value: \(|2x + 1| \geq 2.4\).
02

Create Two Separate Inequalities

Since an absolute value is either positive or negative of its content, the absolute inequality can be interpreted as two separate inequalities: \(2x + 1 \geq 2.4\) and \(2x + 1 \leq -2.4\).
03

Solve each inequality separately

For \(2x + 1 \geq 2.4\), first subtract 1 from both sides: \(2x \geq 1.4\). Then divide both sides by 2 to get \(x \geq 0.7\).For \(2x + 1 \leq -2.4\), first subtract 1 from both sides: \(2x \leq -3.4\). After this, divide both sides by 2 to get \(x \leq -1.7\).

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