Solve symbolically. Express your answer using interval notation -1≤(2-x/3)<(3/5)

To solve the inequality \(-1 \leq \frac{2 - x}{3} < \frac{3}{5}\) symbolically and express the solution in interval notation, follow these steps:

### Step 1: Break the compound inequality into two separate inequalities.

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

### Step 2: Solve the first inequality \(-1 \leq \frac{2 - x}{3}\).

1. Multiply both sides by 3 to eliminate the fraction:
   \[
   -3 \leq 2 - x
   \]

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

3. Multiply by \(-1\) (and reverse the inequality sign):
   \[
   5 \geq x \quad \text{or} \quad x \leq 5
   \]

### Step 3: Solve the second inequality \(\frac{2 - x}{3} < \frac{3}{5}\).

1. Multiply both sides by 3:
   \[
   2 - x < \frac{9}{5}
   \]

2. Subtract 2 from both sides:
   \[
   -x < \frac{9}{5} - 2
   \]

3. Simplify the right side:
   \[
   -x < \frac{9}{5} - \frac{10}{5} = -\frac{1}{5}
   \]

4. Multiply by \(-1\) (and reverse the inequality sign):
   \[
   x > \frac{1}{5}
   \]

### Step 4: Combine the results of both inequalities.

From the first inequality, we have \(x \leq 5\), and from the second inequality, we have \(x > \frac{1}{5}\). Combining these, the solution is:

\[
\frac{1}{5} < x \leq 5
\]

### Step 5: Express the solution in interval notation.

The solution in interval notation is:

\[
\left(\frac{1}{5}, 5\right]
\]

Would you like more details or have any questions?

Here are five related questions:
1. How do you solve inequalities involving absolute values?
2. What happens if we multiply or divide an inequality by a negative number?
3. Can you graph the solution set for this inequality?
4. How would you solve a compound inequality with more than two parts?
5. How does interval notation differ for strict vs. non-strict inequalities?

**Tip:** When solving inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Learn how to solve the compound inequality -1 ≤ (2 - x)/3 < 3/5 symbolically and express the solution using interval notation. This detailed solution includes key algebraic steps, properties of inequalities, and tips for solving compound inequalities. Suitable for students in Grades 8-10.

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