Given:
Which of the following statements is necessarily true?
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Given:
Which of the following statements is necessarily true?
To solve the inequality , we can consider two cases for the absolute value expression.
Case 1:
Rearrange terms:
Case 2:
This simplifies to:
Rearrange terms:
Combining both conditions, the necessary true statement is:
Given:
\( \left|2x-1\right|>-10 \)
Which of the following statements is necessarily true?
Because absolute values are always non-negative! If |2x + 3| ≤ 4x - 2, then 4x - 2 must be ≥ 0. Otherwise, you'd have something ≥ 0 being less than or equal to something < 0, which is impossible.
Case 1: When 2x + 3 ≥ 0, so |2x + 3| = 2x + 3
Case 2: When 2x + 3 < 0, so |2x + 3| = -(2x + 3). You need both cases to cover all possible values of x.
You must satisfy all conditions simultaneously! From Case 1: x ≥ 5/2, from Case 2: x ≥ -1/6, and from the domain: x ≥ 1/2. The intersection of all these is x ≥ 5/2.
After finding the solution, substitute back to check! For x ≥ 5/2, we have 2x + 3 ≥ 8, so 2x + 3 > 0, confirming Case 1 applies to our final answer.
That's possible! If the domain restriction doesn't overlap with either case solution, then the inequality has no solution. Always check this carefully.
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