Given:
Which of the following statements is necessarily true?
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Given:
Which of the following statements is necessarily true?
Firstly, let's analyze the inequality: .
We split it into two separate inequalities:
(1) and (2) .
For inequality (1):
Subtract from both sides:
Subtract 4 from both sides:
Divide both sides by and flip the inequality:
For inequality (2):
Add to both sides:
Subtract 4 from both sides:
Divide both sides by 5:
When we combine the solutions, we see that the answer is
Given:
\( \left|2x-1\right|>-10 \)
Which of the following statements is necessarily true?
The absolute value represents distance from zero, so |2x + 4| could equal either (2x + 4) or -(2x + 4) depending on whether 2x + 4 is positive or negative. You must check both possibilities!
For absolute value inequalities with '>' or '≥', use OR to combine solutions. For '<' or '≤', use AND. Since this problem uses '>', we take the union of both solution sets.
That's normal! Sometimes one case might be impossible or give no valid solutions. Just use the solution from the case that does work. Always check your final answer in the original inequality.
From Case 1 we get x < 3, and from Case 2 we get x < -1. Since we need either condition to be true (OR), the solution includes all values where x < 3, which contains the x < -1 region too.
Pick test values! Try x = 0 (should work since 0 < 3): |2(0) + 4| = 4 and 3(0) + 1 = 1, so 4 > 1 ✓. Try x = 4 (shouldn't work since 4 > 3): |2(4) + 4| = 12 and 3(4) + 1 = 13, so 12 > 13 is false ✓
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