Solving Inequalities

Review how to solve an inequality.

To solve the inequality \( -\frac{3}{4}(x-16) > 18 \), follow these steps:

1. Distribute the coefficient \(-\frac{3}{4}\) across the expression in parentheses:
\[ -\frac{3}{4}(x-16) > 18 \]
\[ -\frac{3}{4}x + \frac{3}{4}(16) > 18 \]

2. Simplify inside the parentheses:
\[ -\frac{3}{4}x + \frac{3}{4}(16) > 18 \]
\[ -\frac{3}{4}x + \frac{3}{4}(16) > 18 \]
\[ -\frac{3}{4}x + 12 > 18 \]

3. Subtract 12 from both sides to isolate the term with \( x \):
\[ -\frac{3}{4}x + 12 - 12 > 18 - 12 \]
\[ -\frac{3}{4}x > 6 \]

4. Multiply both sides by \(-\frac{4}{3}\) to solve for \( x \). Remember to flip the inequality sign when multiplying or dividing by a negative number**:
\[ (-\frac{4}{3}) \times (-\frac{3}{4}x) < (-\frac{4}{3}) \times 6 \]
\[ x < -8 \]

5. Write the solution:
\[ x < -8 \]

So, the solution to the inequality \( -\frac{3}{4}(x-16) > 18 \) is \( x < -8 \).

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