Question

Solve the inequality. Then graph the solution. $$|3 x+2|-1 \geq 10$$

Short answer

The solution to the inequality \(|3x + 2| - 1 \geq 10\) is \(x \leq -\frac{13}{3}\) and \(x \geq 3\).

Step-by-step solution

Isolate the absolute value expression

First, we need to isolate the absolute value expression on one side of the inequality. Rewrite \(|3x + 2| - 1 \geq 10\) as \(|3x + 2| \geq 11\).

Split into two inequalities

The next step is to split this into two separate inequalities without absolute values by considering both negative and positive cases. This gives us \(3x + 2 \geq 11\) and \(3x + 2 \leq -11\).

Solve each inequality

Solve each inequality separately: \n\n For \(3x + 2 \geq 11\), subtract 2 from each side and then divide by 3 to isolate x, yielding \(x \geq 3\). \n\n For \(3x + 2 \leq -11\), subtract 2 from each side and divide by 3 to isolate x, yielding \(x \leq -\frac{13}{3}\).

Graph the solution

Finally, we graph the solution. On a number line, graph \(x \leq -\frac{13}{3}\) to the left side of -\(\frac{13}{3}\) and \(x \geq 3\) to the right side of 3. The solution lies in these two intervals.