Solving Linear Inequalities and Graphing Solution Sets

Q1. Solve the following inequality. Graph the solution set and write it in interval notation: x - 4 ≥ -8

Background

Topic: Linear Inequalities

This question tests your understanding of solving linear inequalities, using properties of inequalities, and representing the solution both graphically and in interval notation.

Key Terms and Formulas:

• Linear Inequality: An inequality involving a linear expression, such as x - 4 ≥ -8.

• Additive Property of Inequality: You can add or subtract the same value from both sides of an inequality without changing the solution set.

• Interval Notation: A way to represent the set of solutions using brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints.

Step-by-Step Guidance

1. Start by identifying the inequality: $x - 4 \geq -8$

2. Apply the addition property of inequality: Add 4 to both sides to isolate $x$.

   $x - 4 + 4 \geq -8 + 4$

3. Simplify both sides: $x \geq -4$

4. Think about how to graph this solution on a number line. Since the inequality is $\geq$, the endpoint $-4$ is included, so use a bracket at $-4$.

5. Shade to the right of $-4$ on the number line, indicating all values greater than or equal to $-4$ are part of the solution set.

   

Try solving on your own before revealing the answer!

Final Answer: The solution in interval notation is $[-4, \infty)$

This interval includes $-4$ (using a bracket) and extends to infinity (using a parenthesis).

The graph shows a bracket at $-4$ and shading to the right, representing all values $x \geq -4$.