
Solve the compound inequality. Use graphs to show the solution set to each of the two given​ inequalities, as well as a third graph that shows the solution set of the compound inequality. Except for the empty​ set, express the solution set in interval notation.
x less than 3 and x greater than or equals minus 6

Graph the solution set of x less than 3. Choose the correct graph below.


We are tasked with solving the compound inequality:

\[
x < 3 \quad \text{and} \quad x \geq -6
\]

### Step 1: Solve Each Inequality Separately

1. **For the first inequality** \(x < 3\), the solution set includes all real numbers that are less than 3. In interval notation, it is expressed as:
   \[
   (-\infty, 3)
   \]
   On a number line, this inequality is represented by an open circle at \(x = 3\), indicating that 3 is not included in the solution, and shading to the left of 3.

2. **For the second inequality** \(x \geq -6\), the solution set includes all real numbers greater than or equal to \(-6\). In interval notation, it is expressed as:
   \[
   [-6, \infty)
   \]
   On a number line, this inequality is represented by a closed circle at \(x = -6\), indicating that \(-6\) is included in the solution, and shading to the right of \(-6\).

### Step 2: Solve the Compound Inequality

The compound inequality requires both conditions to be satisfied simultaneously, meaning we are looking for the intersection of the two solution sets.

- The first inequality \(x < 3\) limits the values to numbers less than 3.
- The second inequality \(x \geq -6\) limits the values to numbers greater than or equal to \(-6\).

The solution set for the compound inequality is the overlap between \((- \infty, 3)\) and \([-6, \infty)\), which results in the solution:

\[
[-6, 3)
\]

In this interval, \(-6\) is included (because \(x \geq -6\)), but 3 is not (because \(x < 3\)).

### Step 3: Graph the Solution

1. **Graph of \(x < 3\)**: An open circle at \(3\) with shading to the left.
2. **Graph of \(x \geq -6\)**: A closed circle at \(-6\) with shading to the right.
3. **Graph of the compound inequality \(x \geq -6 \quad \text{and} \quad x < 3\)**: A closed circle at \(-6\) and an open circle at \(3\), with shading between the two points.

I can create graphs to represent each step if needed! Would you like a detailed visualization? Let me know if you have any questions.

### 5 Related Questions:
1. How do we graph compound inequalities with "or" instead of "and"?
2. What changes in the solution if the inequality is \(x \leq 3\) instead of \(x < 3\)?
3. How can compound inequalities be used in real-world problem-solving?
4. What are the rules for combining inequalities using set intersection and union?
5. How do you solve inequalities with absolute values?

### Tip:
Always remember that inequalities involving "and" require the intersection of solution sets, while "or" requires the union.

Solve the compound inequality. Use graphs to show the solution set to each of the two given inequalities, as well as a third graph that shows the solution set of the compound inequality. Except for the empty set, express the solution set in interval notation. x less than 3 and x greater than or equals minus 6. Graph the solution set of x less than 3. Choose the correct graph below.

This problem focuses on solving the compound inequality x < 3 and x ≥ -6. The solution includes graphing each inequality and finding the intersection of the solution sets. Learn how to express the solution in interval notation and visualize it with graphs. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation