Given:
Which of the following statements is necessarily true?
We have hundreds of course questions with personalized recommendations + Account 100% premium
Given:
Which of the following statements is necessarily true?
To solve the inequality , apply the property of absolute values which states that translates to .
Therefore, .
Subtract 5 from all parts of the inequality to isolate :
This simplifies to .
Given:
\( \left|2x-1\right|>-10 \)
Which of the following statements is necessarily true?
Think of absolute value as distance! |x+5| < 2 means the distance from x to -5 is less than 2. This creates a range of values, so x must satisfy both conditions simultaneously.
Great question! |x+5| > 2 would split into two separate regions: x+5 > 2 OR x+5 < -2. The key difference is < gives you one interval, > gives you two separate intervals.
Remember: absolute value is always positive. So |x+5| < 2 means something positive is less than 2. This creates a bounded interval between two values.
Substitute: |(-4)+5| = |1| = 1. Since 1 < 2 is true, x = -4 is in our solution set. Always check with the original absolute value inequality!
Yes! If you had |x+5| < -1, there would be no solution because absolute values are never negative. But |x+5| < 2 definitely has solutions since 2 is positive.
Get unlimited access to all 18 Absolute Value and Inequality questions, detailed video solutions, and personalized progress tracking.
Unlimited Video Solutions
Step-by-step explanations for every problem
Progress Analytics
Track your mastery across all topics
Ad-Free Learning
Focus on math without distractions
No credit card required • Cancel anytime