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$x$. Remember, since $5$ is positive, the direction of the inequality does not change: $\frac{5x}{5} \geq \frac{10}{5}$.

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 represented by a closed (filled-in) circle at $3$ and shading or an arrow to the left.

4. Compare each graph option to see which one matches this description.

Try solving on your own before revealing the answer!