Problem 31 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$-9 x \geq 36$$

Step-by-Step Solution

Answer

The solution to the inequality \(-9x \geq 36\) is \(x \leq -4\), represented in interval notation as \((-\infty, -4]\).

Step 1: Divide both sides by -9

To isolate x, we need to divide both sides of the inequality by -9. So this will give \(x \leq -4\). Please note, we reversed the inequality symbol because we divided by a negative number.

Step 2: Express the solution in interval notation

The solution \(x \leq -4\) means that x includes every number less than or equal to -4. In interval notation, this is written as \((-\infty, -4]\).

Step 3: Graph the solution on a number line

Lay out a number line, and mark -4 on it. Because the inequality includes -4 (since it's \(x \leq -4\) and not \(x < -4\)), we fill the circle at -4. Draw a line extending to the left from -4 to indicate that all these numbers are included in the solution set.