What is the solution to the inequality |2x + 3| < 7? 4 < x < 10 –5 < x < 2 x < 4 or x > 10
x < –5 or x > 2
The Correct Answer and Explanation is:
We are given the inequality ( |2x + 3| < 7 ) and need to find the solution. Here’s the step-by-step process to solve it:
Step 1: Understanding the Absolute Value Inequality
The absolute value inequality ( |A| < B ) means that the expression inside the absolute value, ( A ), must lie between ( -B ) and ( B ), i.e.,
[
-B < A < B
]
In this case, the expression inside the absolute value is ( 2x + 3 ), and the inequality is ( |2x + 3| < 7 ). Therefore, we can rewrite it as:
[
-7 < 2x + 3 < 7
]
Step 2: Solving the Compound Inequality
Now, we solve the compound inequality ( -7 < 2x + 3 < 7 ).
- Isolate ( 2x ): Subtract 3 from all parts of the inequality:
[
-7 – 3 < 2x < 7 – 3
]
[
-10 < 2x < 4
] - Solve for ( x ): Divide all parts of the inequality by 2:
[
\frac{-10}{2} < x < \frac{4}{2}
]
[
-5 < x < 2
]
Step 3: Conclusion
The solution to the inequality is ( -5 < x < 2 ).
Thus, the correct answer is:
[
\boxed{-5 < x < 2}
]
Explanation:
This solution means that for the absolute value expression ( |2x + 3| ) to be less than 7, ( x ) must be a number between ( -5 ) and ( 2 ). If ( x ) is outside this range (i.e., less than ( -5 ) or greater than ( 2 )), the expression ( 2x + 3 ) would exceed 7 or be less than -7, making the absolute value greater than 7. Hence, the valid values for ( x ) are confined to this interval.