Given:
Which of the following statements is necessarily true?
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Given:
Which of the following statements is necessarily true?
We start with the inequality:
This absolute value inequality breaks into two separate inequalities:
Solving the first inequality:
Subtract from both sides:
Subtract from both sides:
Divide by :
Solving the second inequality:
Distribute the negative:
Add to both sides:
Subtract from both sides:
Divide by :
The solution is the intersection: .
Given:
\( \left|2x-1\right|>-10 \)
Which of the following statements is necessarily true?
The absolute value |expression| represents distance from zero, which can be positive or negative inside. So |2x + 3| > 4x + 5 means either 2x + 3 is positive and greater than 4x + 5, or 2x + 3 is negative but its absolute value is still greater.
You solve both cases and then find where they overlap! The final answer is the intersection of both solutions, not their union.
That's normal! You need to find the intersection of both solution sets. In this problem, x < -4/3 AND x < -1, so the answer is x < -1 (the more restrictive condition).
Yes! If the intersection of both cases is empty, or if the right side is negative (since absolute values are never negative), then there's no solution.
Pick a value from your solution set and substitute it back. For x = -2: and . Since 1 > -3, it works! ✓
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