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

2. Subtract $2$ from both sides to isolate the term with $x$:

   $5x + 2 - 2 \geq 12 - 2$

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

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

   $\frac{5x}{5} \geq \frac{10}{5}$

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 inequality: $x \leq 3$ means all values of $x$ that are less than or equal to $3$.

2. On a number line, a closed (filled-in) circle should be at $3$ to show that $3$ is included in the solution.

3. The arrow should extend to the left from $3$, indicating all values less than $3$ are included.

4. Compare each graph option to see which one matches these criteria.

Try solving on your own before revealing the answer!