Solve the inequalities in Example 2: 2x - 1 2x 2x 2x

$$2x - 1 + 1 \le 7 + 1$$ $$2x \le 8$$

$$\frac{2x}{2} \le \frac{8}{2}$$ $$x \le 4$$

The solution set for this inequality is $\{x : x \le 4\}$.

• Solving the second inequality: $2x \le 8x \le 4$

This is a compound inequality, which can be split into two separate inequalities:

1. $2x \le 8x$
2. $8x \le 4$

Subtract $2x$ from both sides:

$$0 \le 8x - 2x$$ $$0 \le 6x$$

Divide by 6:

$$\frac{0}{6} \le \frac{6x}{6}$$ $$0 \le x \quad \text{or} \quad x \ge 0$$

Divide by 8:

$$\frac{8x}{8} \le \frac{4}{8}$$ $$x \le \frac{1}{2}$$

We have $x \ge 0$ and $x \le \frac{1}{2}$.
Combining these gives:

$$0 \le x \le \frac{1}{2}$$

The solution set for this inequality is $\left\{x : 0 \le x \le \frac{1}{2}\right\}$.

The "Solution set: $\{x : x \le 4\}$" provided in the image corresponds to the solution of the first inequality, $2x - 1 \le 7$.