Solve for inequality and represent it on the number line: $$ -9

Question

Solve for inequality and represent it on the number line: $$ -9<3 x-3 \leq 27 $$ pyright reserved

UpStudy AI · Accepted Answer

1. Start with the inequality:
   $$
   -9 < 3x - 3 \leq 27
   $$

2. Add $$3$$ to all parts of the inequality:
   $$
   -9 + 3 < 3x - 3 + 3 \leq 27 + 3
   $$
   Simplifying, we get:
   $$
   -6 < 3x \leq 30
   $$

3. Divide every part by $$3$$ (since $$3$$ is positive, the inequality signs remain unchanged):
   $$
   \frac{-6}{3} < \frac{3x}{3} \leq \frac{30}{3}
   $$
   Simplifying, we obtain:
   $$
   -2 < x \leq 10
   $$

4. The solution set is:
   $$
   x \in (-2, 10]
   $$

5. Number Line Representation:

- Draw a horizontal line.
   - Place an open circle at $$-2$$ (since $$-2$$ is not included).
   - Place a closed circle at $$10$$ (since $$10$$ is included).
   - Shade the region between $$-2$$ and $$10$$.

The number line is represented as follows:
   
   $$
   \text{---(----○====●----)}
   $$
   
   Here, the open circle at $$-2$$ signifies $$x > -2$$ and the closed circle at $$10$$ signifies $$x \leq 10$$.
The solution to the inequality $$-9 < 3x - 3 \leq 27$$ is $$-2 < x \leq 10$$. On the number line, this is represented by an open circle at $$-2$$ and a closed circle at $$10$$, with the line shaded between them.

Solve for inequality and represent it on the number line: \( -9<3 x-3 \leq 27 \) pyright reserved

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