Question

hw- 2.6 part2 score: 6.65/20 1/2 answered question 2 solve the following inequality, then graph the solution set. |2x - 3| + 1 < 6

Explanation:

Step1: Isolate the absolute value

Subtract 1 from both sides of the inequality \(|2x - 3| + 1 < 6\) to get \(|2x - 3| < 6 - 1\), which simplifies to \(|2x - 3| < 5\).

Step2: Solve the compound inequality

For an absolute value inequality \(|A| < B\) (where \(B>0\)), it is equivalent to \(-B < A < B\). So we have \(-5 < 2x - 3 < 5\).

Step3: Solve for x

First, add 3 to all parts of the compound inequality: \(-5 + 3 < 2x - 3 + 3 < 5 + 3\), which simplifies to \(-2 < 2x < 8\). Then divide all parts by 2: \(\frac{-2}{2} < \frac{2x}{2} < \frac{8}{2}\), resulting in \(-1 < x < 4\).

Answer:

The solution to the inequality \(|2x - 3| + 1 < 6\) is \(-1 < x < 4\). To graph the solution set, we draw an open circle at \(x = -1\) and \(x = 4\) (since the inequality is strict, \(x\) cannot be equal to -1 or 4) and shade the region between -1 and 4 on the number line.