Problem 19 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(-\frac{3}{4} x \geq 30\)

Step-by-Step Solution

Answer

The solution is \( x \leq -40\). The interval notation is \((-\infty, -40]\).

Step 1: Isolate the variable

First, we will isolate the variable by dividing both sides of the inequality by \(-\frac{3}{4}\). To do this, multiply both sides by the reciprocal, which is \(-\frac{4}{3}\). This gives us: \[ x \leq 30 \times -\frac{4}{3} \]

Step 2: Simplify the inequality

Now, simplify the right-hand side of the inequality: \[ x \leq -40 \]

Step 3: Represent the solution on a number line

Graph the solution on a number line. Since the inequality is \( \leq \), use a closed circle at \(-40\), and shade all values to the left: \ <-- shading-->

Step 4: Write the solution in interval notation

Express the solution set using interval notation. Since \ the solution includes \(-40\) and all values less than or equal to \(-40\), the interval is \((-\infty, -40]\)

Key Concepts

Isolate the Variable

When solving inequalities, the first goal is to isolate the variable on one side of the inequality. This means you'll need to get the variable by itself, either on the left or right side. In this particular example, we have the inequality: