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

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

• Isolate $x$: Use algebraic operations to get $x$ by itself on one side of the inequality.

Step-by-Step Guidance

1. Start with the given inequality: $5x + 2 \geq 12$.

2. Subtract $2$from both sides to begin isolating$x$.

3. After subtracting, you will have $5x \geq 10$.

4. Divide both sides by $5$to solve for$x$.

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 the endpoint is included (for $\leq$ or $\geq$).

• Arrow Direction: For $x \leq 3$, shade or draw the arrow to the left of $3$.

Step-by-Step Guidance

1. Identify the point $3$ on the number line.

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

3. Look for a closed (filled-in) circle at $3$to indicate inclusion of $3$.

4. Check which graph has an arrow extending to the left from $3$ (toward negative infinity).

Try solving on your own before revealing the answer!