Question
Solve the following absolute value inequality:
|2x+3|>4
(1 point)
Responses
x<−1 or x>7
x<−1 or x>7
x<43 or x>−32
x<43 or x>−32
x<−72 or x>12
x<−72 or x>12
x<−7 or x>1
|2x+3|>4
(1 point)
Responses
x<−1 or x>7
x<−1 or x>7
x<43 or x>−32
x<43 or x>−32
x<−72 or x>12
x<−72 or x>12
x<−7 or x>1
Answers
GPT-4o mini
To solve the absolute value inequality \(|2x + 3| > 4\), we need to consider the two cases for the absolute value.
1. **Case 1**: \(2x + 3 > 4\)
\[
2x + 3 > 4 \\
2x > 4 - 3 \\
2x > 1 \\
x > \frac{1}{2}
\]
2. **Case 2**: \(2x + 3 < -4\)
\[
2x + 3 < -4 \\
2x < -4 - 3 \\
2x < -7 \\
x < -\frac{7}{2}
\]
Combining both cases, we have the solution:
\[
x < -\frac{7}{2} \quad \text{or} \quad x > \frac{1}{2}
\]
Now, looking at the provided options, we can find which of these corresponds to our solution.
The solutions can also be expressed as:
- \(x < -3.5\) (this can be interpreted as \(x < -3.5\) or \(x < -7/2\))
- \(x > 0.5\) (this can be interpreted as \(x > 0.5\) or \(x > 1/2\))
Now we check the options given in the responses:
1. **\(x < -1\) or \(x > 7\)**: This does not match our bounds.
2. **\(x < 43\) or \(x > -32\)**: This is always true for real numbers.
3. **\(x < -72\) or \(x > 12\)**: This does not match our bounds.
4. **\(x < -7\) or \(x > 1\)**: This moves too far to the left and right.
To summarize, none of the provided responses match the exact solution to the inequality \(|2x + 3| > 4\).
Thus, the answer is:
- \(x < -3.5\) (or \(x < -\frac{7}{2}\)) and \(x > \frac{1}{2}\) (or approximately \(x > 0.5\)).
Since these values do not appear in any of the options provided, please verify the answers given to ensure they are correct as per the requirements we have.
1. **Case 1**: \(2x + 3 > 4\)
\[
2x + 3 > 4 \\
2x > 4 - 3 \\
2x > 1 \\
x > \frac{1}{2}
\]
2. **Case 2**: \(2x + 3 < -4\)
\[
2x + 3 < -4 \\
2x < -4 - 3 \\
2x < -7 \\
x < -\frac{7}{2}
\]
Combining both cases, we have the solution:
\[
x < -\frac{7}{2} \quad \text{or} \quad x > \frac{1}{2}
\]
Now, looking at the provided options, we can find which of these corresponds to our solution.
The solutions can also be expressed as:
- \(x < -3.5\) (this can be interpreted as \(x < -3.5\) or \(x < -7/2\))
- \(x > 0.5\) (this can be interpreted as \(x > 0.5\) or \(x > 1/2\))
Now we check the options given in the responses:
1. **\(x < -1\) or \(x > 7\)**: This does not match our bounds.
2. **\(x < 43\) or \(x > -32\)**: This is always true for real numbers.
3. **\(x < -72\) or \(x > 12\)**: This does not match our bounds.
4. **\(x < -7\) or \(x > 1\)**: This moves too far to the left and right.
To summarize, none of the provided responses match the exact solution to the inequality \(|2x + 3| > 4\).
Thus, the answer is:
- \(x < -3.5\) (or \(x < -\frac{7}{2}\)) and \(x > \frac{1}{2}\) (or approximately \(x > 0.5\)).
Since these values do not appear in any of the options provided, please verify the answers given to ensure they are correct as per the requirements we have.