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Chapter 0: Problem 62

Solve each absolute value inequality. $$|3 x+5|<17$$

Short Answer

Expert verified
The solution to the absolute value inequality |3x+5|<17 is \(-22/3 < x < 4\), or in decimal representation \(-7.33 < x < 4\).

Step by step solution

01

Write the Equivalent Inequalities

An inequality with absolute values can be written as two separate inequalities. In this case, |3x+5|<17 translates to -17 < 3x+5 < 17.
02

Solve the First Inequality

To solve -17 < 3x+5 start by subtracting 5 from all parts of the inequality: -17-5 < 3x+5-5 transforms into -22 < 3x, then divide every part of the inequality by 3: \(-22/3 < x\)
03

Solve the Second Inequality

To solve 3x+5 < 17 start by subtracting 5 from both sides: 3x+5-5 < 17-5 transforms into 3x < 12, then divide both sides of the inequality by 3: \(x < 4\)
04

Keeping Together both Inequalities

Since both inequalities are true simultaneously, the solution is the intersection of the solution sets of both inequalities, therefore the final solution, taking into account both inequalities, is \(-22/3 < x < 4\) or using decimals: \(-7.33 < x < 4\)

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Most popular questions from this chapter

Perform the indicated operations. Simplify the result, if possible. $$\left(\frac{1}{a^{3}-b^{3}} \cdot \frac{a c+a d-b c-b d}{1}\right)-\frac{c-d}{a^{2}+a b+b^{2}}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can check inequalities by substituting 0 for the variable: When 0 belongs to the solution set, I should obtain a true statement, and when 0 does not belong to the solution set. I should obtain a false statement.

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