Solving Linear Inequalities and Graphing Solutions on a Number Line

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

Background

Topic: Linear Inequalities

This question tests your ability to solve a linear inequality for $x$ and interpret the solution in set notation.

Key Terms and Formulas:

• Inequality: A mathematical statement that relates expressions that are not necessarily equal, using symbols like $\geq$, $\leq$, $>$, or $<$.

• Solving an Inequality: Similar to solving equations, but you must be careful with the direction of the inequality, especially when multiplying or dividing by a negative number.

Step-by-Step Guidance

1. Start by isolating the term with $x$ on one side. Subtract $2$ from both sides of the inequality: $5x + 2 - 2 \geq 12 - 2$.

2. Simplify both sides to get $5x \geq 10$.

3. Divide both sides by $5$ to solve for $x$: $\frac{5x}{5} \geq \frac{10}{5}$.

4. Simplify the result to get $x \geq$ (complete this step).

Try solving on your own before revealing the answer!

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:

• Closed Circle: Used on a number line to indicate that a number is included in the solution set (for $\leq$ or $\geq$).

• Arrow Direction: Indicates all numbers less than or greater than a certain value.

Step-by-Step Guidance

1. Identify the value where the inequality changes: $x = 3$.

2. Since the inequality is $x \leq 3$, the solution includes $3$ and all numbers less than $3$.

3. On the number line, this is shown with a closed (filled-in) circle at $3$ and shading or an arrow to the left.

4. Examine each graph option and look for the one that matches these features.

Try solving on your own before revealing the answer!