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 split it into two separate inequalities:
or .
For the first inequality , subtract 1 from both sides:
Divide by 2:
For the second inequality , subtract 1 from both sides:
Divide by 2:
Therefore, the solution is or .
or
Given:
\( \left|2x-1\right|>-10 \)
Which of the following statements is necessarily true?
The absolute value measures distance from zero, so means the expression inside is either more than 7 units positive OR more than 7 units negative from zero.
For absolute value inequalities with greater than (>), always use OR. The solution is two separate regions on the number line. For less than (<), you'd use AND because the solution would be between two values.
That would be the solution to ! Since we have greater than 7, we want values that make the absolute value large, which happens at the extremes of the number line.
Substitute: . Since 9 > 7, x = 4 is in our solution set! Try x = 0: , so it's not a solution.
Draw two arrows: one pointing left from -4 (not including -4) and one pointing right from 3 (not including 3). The solution includes all numbers in these two regions.
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