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?

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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.