Question
Solve the inequality. Graph and check your solutions.
one seventh plus a less than or equals minus six sevenths
Question content area bottom
Part 1
a
▼
less than
greater than or equals
greater than
less than or equals
enter your response here
(Simplify your answer.)
one seventh plus a less than or equals minus six sevenths
Question content area bottom
Part 1
a
▼
less than
greater than or equals
greater than
less than or equals
enter your response here
(Simplify your answer.)
Answers
GPT-4o mini
To solve the inequality \( \frac{1}{7} + a \leq -\frac{6}{7} \), we can follow these steps:
1. **Isolate \( a \)**: Subtract \( \frac{1}{7} \) from both sides of the inequality.
\[
a \leq -\frac{6}{7} - \frac{1}{7}
\]
2. **Combine the fractions**:
- Since both fractions have a denominator of 7, we can combine them:
\[
-\frac{6}{7} - \frac{1}{7} = -\frac{6 + 1}{7} = -\frac{7}{7} = -1
\]
3. **Write the final inequality**:
\[
a \leq -1
\]
So the solution to the inequality is:
\[
a \leq -1
\]
### Part 2: Graphing the Solution
To graph the solution \( a \leq -1 \):
- On a number line, place a solid dot at -1 to indicate that -1 is included in the solution.
- Shade the line to the left of -1 to represent all numbers less than -1.
### Part 3: Check the Solution
To check if our solution is correct:
1. **Choose a value less than -1**, like -2:
\[
\frac{1}{7} + (-2) = \frac{1}{7} - \frac{14}{7} = -\frac{13}{7}
\]
Check if \( -\frac{13}{7} \leq -\frac{6}{7} \): This is true.
2. **Choose a value greater than -1**, like 0:
\[
\frac{1}{7} + 0 = \frac{1}{7}
\]
Check if \( \frac{1}{7} \leq -\frac{6}{7} \): This is false.
Since our check confirms that any number \( a \) that is less than or equal to -1 satisfies the original inequality, we have verified that the solution \( a \leq -1 \) is correct.
1. **Isolate \( a \)**: Subtract \( \frac{1}{7} \) from both sides of the inequality.
\[
a \leq -\frac{6}{7} - \frac{1}{7}
\]
2. **Combine the fractions**:
- Since both fractions have a denominator of 7, we can combine them:
\[
-\frac{6}{7} - \frac{1}{7} = -\frac{6 + 1}{7} = -\frac{7}{7} = -1
\]
3. **Write the final inequality**:
\[
a \leq -1
\]
So the solution to the inequality is:
\[
a \leq -1
\]
### Part 2: Graphing the Solution
To graph the solution \( a \leq -1 \):
- On a number line, place a solid dot at -1 to indicate that -1 is included in the solution.
- Shade the line to the left of -1 to represent all numbers less than -1.
### Part 3: Check the Solution
To check if our solution is correct:
1. **Choose a value less than -1**, like -2:
\[
\frac{1}{7} + (-2) = \frac{1}{7} - \frac{14}{7} = -\frac{13}{7}
\]
Check if \( -\frac{13}{7} \leq -\frac{6}{7} \): This is true.
2. **Choose a value greater than -1**, like 0:
\[
\frac{1}{7} + 0 = \frac{1}{7}
\]
Check if \( \frac{1}{7} \leq -\frac{6}{7} \): This is false.
Since our check confirms that any number \( a \) that is less than or equal to -1 satisfies the original inequality, we have verified that the solution \( a \leq -1 \) is correct.