Problem 64 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution for the absolute value inequality \(|3x + 5| < 17\) is \(-7.33 < x < 4\).

Step 1: Isolate the absolute value

The inequality from the exercise is \(|3 x+5|<17\). To isolate the absolute value on one side of the inequality, the equation remains the same.

Step 2: Splitting into two inequalities

Now split the absolute value inequality into two separate inequalities: \(3x + 5 < 17\) and \(3x + 5 > -17\)

Step 3: Simplify both inequalities

For \(3x + 5 < 17\), subtract 5 from both sides to get \(3x < 12\). Then divide both sides by 3 to get \(x < 4\).\nFor \(3x + 5 > -17\), subtract 5 from both sides to get \(3x > -22\). Then divide both sides by 3 to get \(x > -22/3\) which is equivalent to \(x > -7.33\) when rounded to two decimal places.

Step 4: Combine the solutions

Finally, the solution to the original absolute value inequality \(|3 x+5|<17\) is from step 3, which is \(x < 4\) and \(x > -7.33\) or in interval notation, \(-7.33 < x < 4\).