Question

Solve the inequality. Then sketch a graph of the solution on a number line. $$|3 x+7|-4>9 \quad $$

Short answer

The solution of the inequality is \( x > 2 \) or \( x < -\frac{20}{3} \).

Step-by-step solution

Isolate the absolute value expression

Add 4 to both sides of the inequality to isolate the absolute value term: |3x + 7| > 13.

Rewrite as compound inequality

Now, rewrite the absolute value inequality as a compound inequality without absolute value symbols: either 3x + 7 > 13 or 3x + 7 < -13.

Solve the compound inequality

For 3x + 7 > 13, subtract 7 from both sides and then divide by 3, we get: x > 2. For 3x + 7 < -13, subtract 7 from both sides and then divide by 3, we get: x < -20/3.

Represent the solution on a number line

Draw a number line and mark points -20/3 and 2. The solution to the compound inequality is x > 2 or x < -20/3. This means solution includes all numbers greater than 2 and less than -20/3.