Given:
Which of the following statements is necessarily true?
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Given:
Which of the following statements is necessarily true?
The absolute value expression inherently suggests that for any real number , the inequality holds.
Since the absolute value of any expression is always non-negative and is negative, the condition is always satisfied regardless of the choice of .
Thus, there is no specific limitation or exceptional circumstance that confines to any particular subset of the real numbers.
This implies that no particular statement about being greater, less, or constrained to a specific domain can be justified. Therefore, the notion of any statement being "necessarily true" in the conventional sense of constraining does not apply.
The correct answer, therefore, is: all .
all
Given:
\( \left|2x-1\right|>-10 \)
Which of the following statements is necessarily true?
The absolute value of any expression is always zero or positive. Since and , we always have !
That would have no solution! Since absolute values are never negative, can never be less than -11. The solution set would be empty.
With , you need to solve two cases: x-5 > 11 or x-5 < -11, giving x > 16 or x < -6. But with a negative number on the right, all values work!
Never ignore absolute value signs! They're crucial for understanding the problem. In this case, they help us recognize that we're comparing a non-negative value to a negative number.
Pick any value for x and substitute it in! Try x = 0: ✓. Try x = 100: ✓. Every value works!
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