Solve the compound inequality and write the solution in interval notation:  
3/
4
x
−
3
≤
3
  or  
2/
5
(
x
+
10
)
≥
0.

Let's break down the compound inequality and solve it step by step.

The given inequalities are:

1. \(\frac{3}{4}x - 3 \leq 3\)
2. \(\frac{2}{5}(x + 10) \geq 0\)

### Step 1: Solve the first inequality
\[
\frac{3}{4}x - 3 \leq 3
\]
Add 3 to both sides:
\[
\frac{3}{4}x \leq 6
\]
Now, multiply both sides by \(\frac{4}{3}\) to isolate \(x\):
\[
x \leq 6 \times \frac{4}{3}
\]
\[
x \leq 8
\]

### Step 2: Solve the second inequality
\[
\frac{2}{5}(x + 10) \geq 0
\]
Multiply both sides by 5 to get rid of the denominator:
\[
2(x + 10) \geq 0
\]
Divide both sides by 2:
\[
x + 10 \geq 0
\]
Subtract 10 from both sides:
\[
x \geq -10
\]

### Step 3: Combine the solutions
The compound inequality involves the word "or", so we combine the solutions as follows:

- From the first inequality: \(x \leq 8\)
- From the second inequality: \(x \geq -10\)

Since this is an "or" inequality, the solution includes any value that satisfies either of the inequalities. Therefore, the solution is all real numbers between \(-10\) and 8, inclusive.

### Final Solution in Interval Notation:
\[
[-10, 8]
\]

Would you like more details on any step, or have any further questions?

---

Here are 5 related questions for practice:

1. Solve and write in interval notation: \(2x - 5 \geq 3 \text{ or } 3x + 4 \leq 10\).
2. Solve \(5x + 2 \leq 7 \text{ and } 4x - 3 \geq 1\) in interval notation.
3. Solve \(x^2 - 5x \leq 0\) and express the solution in interval notation.
4. How do we combine inequalities using "and" vs "or" in interval notation?
5. Solve and write the solution for \(\frac{1}{2}(x - 4) > 3 \text{ or } \frac{3}{4}(x + 2) < 5\).

**Tip:** In compound inequalities, the word "or" allows a solution to satisfy either inequality, while "and" requires it to satisfy both.

Solve the compound inequality and write the solution in interval notation: 3/4x − 3 ≤ 3 or 2/5(x + 10) ≥ 0.

Master the steps to solve the compound inequality 3/4x − 3 ≤ 3 or 2/5(x + 10) ≥ 0. Learn how to isolate the variable, combine solutions using 'or', and express the final result in interval notation. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation