Solving Compound Inequalities in Intermediate Algebra

Q1. Solve the compound inequality: $6x + 3 \leq -9$ and $5x \leq 0$

Background

Topic: Compound Inequalities

This question tests your ability to solve compound inequalities by isolating the variable in each inequality and then finding the intersection of their solution sets.

Key Terms and Formulas

• Compound Inequality: An inequality that combines two or more inequalities, often using "and" or "or".

• Interval Notation: A way to represent the set of solutions using intervals.

• Properties of Inequalities: You can add, subtract, multiply, or divide both sides of an inequality by the same number (except when multiplying/dividing by a negative, which reverses the inequality).

Step-by-Step Guidance

1. Start by solving the first inequality: $6x + 3 \leq -9$.

2. Subtract 3 from both sides to isolate the $x$ term: $6x + 3 - 3 \leq -9 - 3$.

3. Combine like terms: $6x \leq -12$.

4. Divide both sides by 6 to solve for $x$: $\frac{6x}{6} \leq \frac{-12}{6}$.

5. Now, solve the second inequality: $5x \leq 0$.

6. Divide both sides by 5: $\frac{5x}{5} \leq \frac{0}{5}$.

Try solving on your own before revealing the answer!

Final Answer: $(-\infty, -2]$

The solution set is the intersection of $x \leq -2$ and $x \leq 0$, which is $x \leq -2$. In interval notation, this is $(-\infty, -2]$.

This represents all values of $x$ that satisfy both inequalities simultaneously.