Solve - 5 ≤ 5-3x/2 ≤8 and draw number line

To solve the compound inequality \( -5 \leq \frac{5 - 3x}{2} \leq 8 \), we will break it into two parts:

1. \( -5 \leq \frac{5 - 3x}{2} \)
2. \( \frac{5 - 3x}{2} \leq 8 \)

### Part 1: Solve \( -5 \leq \frac{5 - 3x}{2} \)

1. Multiply all parts by 2 to eliminate the denominator:
   \[
   -10 \leq 5 - 3x
   \]
   
2. Subtract 5 from both sides:
   \[
   -15 \leq -3x
   \]

3. Divide by -3 (remember, dividing by a negative reverses the inequality sign):
   \[
   5 \geq x \quad \text{or} \quad x \leq 5
   \]

### Part 2: Solve \( \frac{5 - 3x}{2} \leq 8 \)

1. Multiply all parts by 2:
   \[
   5 - 3x \leq 16
   \]

2. Subtract 5 from both sides:
   \[
   -3x \leq 11
   \]

3. Divide by -3 (reverse the inequality sign):
   \[
   x \geq -\frac{11}{3}
   \]

### Combine the results

The solution is \( -\frac{11}{3} \leq x \leq 5 \).

### Number Line Representation

The number line will have a closed circle at \( x = -\frac{11}{3} \) and \( x = 5 \), with shading between them.

I can generate a number line graph for this solution if you'd like. Would you like me to generate it?

### Related Questions:

1. How do you solve other compound inequalities involving fractions?
2. What happens if you multiply both sides of an inequality by a negative number?
3. How would the solution change if it was \( - 5 < 5 - \frac{3x}{2} < 8 \)?
4. Can inequalities have no solution? In what cases?
5. How do absolute value inequalities work?

**Tip:** Always reverse the inequality sign when multiplying or dividing by a negative number in inequalities.

Solve -5 ≤ (5 - 3x)/2 ≤ 8 and draw number line

Learn how to solve the compound inequality -5 ≤ (5 - 3x)/2 ≤ 8 step-by-step. This problem covers key algebraic concepts like solving inequalities, handling compound inequalities, and representing solutions on a number line. Suitable for students in Grades 9-10.

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