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 , follow these steps:
Step 1: Remove the absolute value by expressing it as a double inequality:
.
Step 2: Simplify the inequality:
First, subtract 9 from all parts:
,
which simplifies to .
Step 3: Solve for by dividing the entire inequality by 3:
,
resulting in .
Upon solving, we determine that the solution to the inequality is the interval:
.
Given:
\( \left|2x-1\right|>-10 \)
Which of the following statements is necessarily true?
The absolute value |A| represents distance from zero. When |A| < 18, the expression A must be between -18 and 18, giving us the compound inequality -18 < A < 18.
With |A| < B, you get a compound inequality (AND condition): -B < A < B. With |A| > B, you get two separate inequalities (OR condition): A < -B or A > B.
Pick any number inside your interval and substitute it back. For , try x = 0: |3(0) + 9| = 9, and 9 < 18 ✓
The original inequality uses strict inequality (<). This means the absolute value must be strictly less than 18, not equal to it. So the endpoints are not included.
Work systematically: Start with -18 < 3x + 9 < 18, then subtract 9 from all three parts, then divide all three parts by 3. Keep the inequality signs pointing the same direction!
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