Problem 42 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$\frac{3 x}{10}+1 \geq \frac{1}{5}-\frac{x}{10}$$

Step-by-Step Solution

Answer

The solution to the inequality \(\frac{3x}{10} + 1 \geq \frac{1}{5} - \frac{x}{10}\) is \(x \geq -2\). In interval notation, this is \([-2, \infty)\). The graph on a number line starts at -2 (included) and extends to the right to represent all numbers greater than -2.

Step 1: Isolate the variable \(x\)

First, let's get all terms with \(x\) on one side of the inequality and the constants on the other side: \(\frac{3 x}{10}+\frac{x}{10} \geq \frac{1}{5}-1\). Adding like terms gives: \(\frac{4x}{10} \geq \frac{1}{5} - 1\). Multiply both sides by 10: \(4x \geq 2 - 10\). Thus, we obtain: \(4x \geq -8\).

Step 2: Solve for \(x\)

Now, isolate the \(x\) by dividing both sides by 4. This gives: \(x \geq -2\). This is the solution for \(x\) in inequality form.

Step 3: Write solution in interval notation

The solution in interval notation is \([-2, \infty)\). This includes all real numbers greater than or equal to -2.

Step 4: Graph the solution set

On a number line, a closed circle is used to indicate that -2 is included in the solution set, and an arrow extending to the right to represent all values greater than -2.