Question

Solve the inequality. Then graph the solution. $$|3 x-9|+2>7$$

Short answer

The solution to the inequality is \(x< \frac{4}{3}\) or \(x> \frac{14}{3}\).

Step-by-step solution

Simplify the Inequality

First, subtract 2 from both sides of the inequality to isolate the absolute value: \(|3x-9| > 7 - 2\), which simplifies to \(|3x-9| > 5)

Consider the Two Cases

Now, split the inequality into two cases: 3x - 9 > 5 (Case 1) and -(3x - 9) > 5 (Case 2)

Solve Case 1

Case 1: 3x - 9 > 5. Add 9 to both sides: 3x > 5 + 9, which simplifies to 3x > 14. Then divide both sides by 3 to solve for x, getting x > 14/3.

Solve Case 2

Case 2: -(3x - 9) > 5. Distribute the negative sign inside the parenthesis: -3x + 9 > 5. Then, subtract 9 from both sides: -3x > 5 - 9, which simplifies to -3x > -4. Dividing by -3 (remember to flip the sign of inequality when multiplied or divided by negative number) gives x < 4/3.

Graph the Solution

On the number line, shade the region to the right of 14/3 and the region to the left of 4/3.