College Algebra Study Guidance: Solving Inequalities, Graphing Solutions, and Simplifying Expressions

Q9. Solve: $5x + 2 \geq 12$

Background

Topic: Solving Linear Inequalities

This question tests your ability to solve a linear inequality and interpret the solution in terms of the variable $x$.

Key Terms and Formulas:

• Linear inequality: An inequality involving a linear expression, such as $ax + b \geq c$.

• Solving inequalities: Similar to solving equations, but pay attention to the direction of the inequality sign.

Step-by-Step Guidance

1. $\text{Start by isolating the variable } x \text{ in the inequality: } 5x + 2 \geq 12$

2. $\text{Subtract 2 from both sides to get: } 5x \geq 10$

3. $\text{Divide both sides by 5 (since 5 is positive, the inequality direction stays the same): } x \geq 2$

Try solving on your own before revealing the answer!

Final Answer: $x \geq 2$

This means all values of $x$ greater than or equal to 2 satisfy the inequality.

Q10. Which answer choice below is the graph of $x \leq 3$?

Background

Topic: Graphing Solutions to Inequalities on a Number Line

This question tests your ability to interpret and graph the solution set of an inequality on a number line.

Key Terms and Concepts:

• Number line: A visual representation of numbers in order.

• Closed circle: Indicates the endpoint is included (e.g., $x \leq 3$ includes 3).

• Arrow: Shows the direction of the solution set (left for $\leq$, right for $\geq$).

Step-by-Step Guidance

1. $\text{Identify the inequality: } x \leq 3$

2. $\text{On a number line, this means all values to the left of 3, including 3 itself.}$

3. $\text{Look for a graph with a closed circle at 3 and shading (arrow) to the left.}$

Try solving on your own before revealing the answer!

Final Answer: Choice a

Choice a shows a closed circle at 3 and shading to the left, which matches $x \leq 3$.

Q11. Simplify: $3x + x + 5x + 2x + x$

Background

Topic: Combining Like Terms

This question tests your ability to simplify algebraic expressions by combining like terms.

Key Terms and Concepts:

• Like terms: Terms that have the same variable raised to the same power (e.g., $x$ terms).

• Simplifying: Adding the coefficients of like terms together.

Step-by-Step Guidance

1. $\text{List all the coefficients of } x: 3, 1, 5, 2, 1$

2. $\text{Add the coefficients together: } 3 + 1 + 5 + 2 + 1$

3. $\text{Write the simplified expression as } (\text{sum})x$

Try solving on your own before revealing the answer!

Final Answer: $12x$

Adding all the coefficients gives $12$, so the simplified expression is $12x$.