Given:
Which of the following statements is necessarily true?
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Given:
Which of the following statements is necessarily true?
To solve , we need to consider two cases based on the definition of absolute value:
1.
2.
Solving the first inequality:
Solving the second inequality:
These inequalities indicate that is the range that satisfies the original inequality.
Given:
\( \left|2x-1\right|>-10 \)
Which of the following statements is necessarily true?
The absolute value |expression| means the distance from zero, which is always positive. So |2x + 3| could equal either (2x + 3) when it's positive, or -(2x + 3) when it's negative!
You don't choose! You solve both cases and then combine the results. The final answer includes all x-values that satisfy either case.
This case handles when the expression inside the absolute value bars (2x + 3) is negative. We flip it to positive by multiplying by -1, so |2x + 3| becomes -(2x + 3).
Great question! When we solve case 1, we get x < 1. Case 2 gives x < -2/3. Since every number less than -2/3 is also less than 1, the broader condition x < 1 includes both cases.
Pick a test value! Try x = 0: and . Since 3 > 1, our solution x < 1 works! ✓
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