Question
solve and graph the solution to the inequality (-6n+3 greater than or equal to 21). then explain what the solution means.
Answers
GPT-4o mini
To solve the inequality \(-6n + 3 \geq 21\), we will follow these steps:
1. **Isolate the variable term**:
\[
-6n + 3 \geq 21
\]
Subtract 3 from both sides:
\[
-6n \geq 21 - 3
\]
\[
-6n \geq 18
\]
2. **Divide both sides by -6**:
Note: When dividing or multiplying both sides of an inequality by a negative number, the inequality sign reverses.
\[
n \leq \frac{18}{-6}
\]
\[
n \leq -3
\]
### Solution:
The solution to the inequality is:
\[
n \leq -3
\]
### Graphing the Solution:
To graph the solution \(n \leq -3\) on a number line:
1. Draw a number line.
2. Locate \(-3\) on the number line.
3. Since the inequality is "less than or equal to," represent \(-3\) with a closed circle (indicating that \(-3\) is included in the solution).
4. Shade the region to the left of \(-3\) to show all values that are less than \(-3\).
### Explanation of the Solution:
The solution \(n \leq -3\) means that any value of \(n\) that is less than or equal to \(-3\) satisfies the original inequality. In practical terms, if you were to substitute any number less than or equal to \(-3\) back into the original inequality \(-6n + 3 \geq 21\), the inequality would hold true. For example, \(n = -4\) yields \(-6(-4) + 3 = 24 + 3 = 27\), which is greater than 21. Thus, this indicates a range of possible values for \(n\) that satisfy the condition given in the inequality.
1. **Isolate the variable term**:
\[
-6n + 3 \geq 21
\]
Subtract 3 from both sides:
\[
-6n \geq 21 - 3
\]
\[
-6n \geq 18
\]
2. **Divide both sides by -6**:
Note: When dividing or multiplying both sides of an inequality by a negative number, the inequality sign reverses.
\[
n \leq \frac{18}{-6}
\]
\[
n \leq -3
\]
### Solution:
The solution to the inequality is:
\[
n \leq -3
\]
### Graphing the Solution:
To graph the solution \(n \leq -3\) on a number line:
1. Draw a number line.
2. Locate \(-3\) on the number line.
3. Since the inequality is "less than or equal to," represent \(-3\) with a closed circle (indicating that \(-3\) is included in the solution).
4. Shade the region to the left of \(-3\) to show all values that are less than \(-3\).
### Explanation of the Solution:
The solution \(n \leq -3\) means that any value of \(n\) that is less than or equal to \(-3\) satisfies the original inequality. In practical terms, if you were to substitute any number less than or equal to \(-3\) back into the original inequality \(-6n + 3 \geq 21\), the inequality would hold true. For example, \(n = -4\) yields \(-6(-4) + 3 = 24 + 3 = 27\), which is greater than 21. Thus, this indicates a range of possible values for \(n\) that satisfy the condition given in the inequality.