Question

Solve the inequality. Then graph the solution. $$|5 x+3|-4 \geq 9$$

Short answer

The solution to the inequality is \(x \leq -16/5\) and \(x \geq 2\).

Step-by-step solution

Isolate The Absolute Value

Start by isolating the absolute value. This is done by adding 4 to both sides of the inequality, resulting in \(|5x + 3| \geq 13\).

Solve When The Expression Is Positive

When the expression inside the absolute value is positive, we rewrite the inequality without the absolute value, \(5x + 3 \geq 13\). We then solve the inequality by subtracting 3 from both sides and dividing by 5, giving \(x \geq 2\).

Solve When The Expression Is Negative

When the expression inside the absolute value is negative, we change the direction of the inequality and multiply the right side by -1, giving \(-(5x + 3) \geq 13\). Simplifying this gives \(x \leq -16/5\).

Graph The Solution

On a number line, graph \(x \geq 2\) with a closed circle at 2 and shading to the right. Also, graph \(x \leq -16/5\) with a closed circle at -16/5 and shading to the left.