Solve the inequality for $$ x $$ $$ -\frac{x}{8} \leq-\frac{3}{4} $$

Question

UpStudy AI · Accepted Answer

To solve the inequality:

$$
-\frac{x}{8} \leq -\frac{3}{4}
$$

follow these steps:

### Step 1: Eliminate the Negative Sign
First, multiply both sides of the inequality by $$-1$$. **Important:** When you multiply or divide both sides of an inequality by a negative number, you must **reverse the inequality sign**.

$$
-\frac{x}{8} \leq -\frac{3}{4} \quad \xrightarrow{\times (-1)} \quad \frac{x}{8} \geq \frac{3}{4}
$$

### Step 2: Solve for $$ x $$
Now, isolate $$ x $$ by multiplying both sides by $$ 8 $$:

$$
\frac{x}{8} \geq \frac{3}{4} \quad \xrightarrow{\times 8} \quad x \geq \frac{3}{4} \times 8
$$

Calculate the right side:

$$
x \geq \frac{3}{4} \times 8 = 6
$$

### Final Solution
$$
x \geq 6
$$

**Therefore, all real numbers $$ x $$ that are greater than or equal to $$ 6 $$ satisfy the inequality.**

### Graphical Representation
On a number line, the solution is represented by a closed circle at $$ 6 $$ and a shaded region extending to the right:

```
<---|----|----|----|----|----|---->
    5    6    7    8    9   10
         ●================>
```

Here, the closed circle at $$ 6 $$ indicates that $$ 6 $$ is included in the solution set, and the arrow pointing to the right signifies all numbers greater than $$ 6 $$.
$$ x \geq 6 $$

Solve the inequality for \( x \) \[ -\frac{x}{8} \leq-\frac{3}{4} \]

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