Given:\( |3c + 5| + |-c - 6| Which of the following statements is necessarily true?

This equation is valid for some $ c $

Solve Double Absolute Value Inequality: |3c + 5| + |-c - 6| < -1

Q: Why can't absolute values be negative?

By definition, absolute value measures distance from zero, which is never negative. So $ |x| \ge 0 $ for any real number x.

Q: What if one absolute value could be negative to cancel the other?

Impossible! Both $ |3c + 5| \ge 0 $ and $ |-c - 6| \ge 0 $ are always non-negative, so their sum is always ≥ 0.

Q: How do I recognize when an inequality has no solution?

Look for situations where you need a positive quantity to be negative. Examples: \( x^2 , \( |x| , or sums of absolute values less than negative numbers.

Q: Should I still try to solve it step by step?

No need! Once you recognize that $ |a| + |b| \ge 0 $ but the inequality requires , you can immediately conclude no solution exists.

Q: What's the difference between 'no solution' and 'all real numbers'?

No solution: The inequality is impossible (like our problem). All real numbers: Every value works (like $ |x| \ge 0 $).

Absolute Value Inequalities with Impossible Conditions

Given:

$|3c + 5| + |-c - 6| < -1$

Which of the following statements is necessarily true?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given:

$|3c + 5| + |-c - 6| < -1$

Which of the following statements is necessarily true?

Step-by-step solution

The given inequality is: $|3c + 5| + |-c - 6| < -1$ .

Combining absolute values with negative numbers results in an inequality that cannot be less than $-1$ .

To show this, consider each term separately: both $|3c + 5| \ge 0$ and $|-c - 6| \ge 0$ because absolute values cannot be negative.

Add these terms: $|3c + 5| + |-c - 6| \ge 0$ . Clearly, this result cannot be less than -1.

Therefore, the condition $< -1$ cannot be satisfied for any $c$ .

Thus, the statement "No solution" is correct.

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Fundamental Rule: Sum of absolute values is always non-negative
Technique: Check if $|a| + |b| \ge 0$ can satisfy given inequality
Check: When sum ≥ 0 but inequality requires < -1, no solution exists ✓

Common Mistakes

Avoid these frequent errors

Attempting to solve by finding critical points
Don't break down $|3c + 5| + |-c - 6| < -1$ into cases and solve each = wasted time on impossible problem! Since absolute values are never negative, their sum cannot be less than -1. Always check if the inequality is mathematically possible before solving.

Practice Quiz

Test your knowledge with interactive questions

Find the absolute value inequality representation for:

$|x + 3| \leq 5$

FAQ

Everything you need to know about this question

Why can't absolute values be negative?

By definition, absolute value measures distance from zero, which is never negative. So $|x| \ge 0$ for any real number x.

What if one absolute value could be negative to cancel the other?

Impossible! Both $|3c + 5| \ge 0$ and $|-c - 6| \ge 0$ are always non-negative, so their sum is always ≥ 0.

How do I recognize when an inequality has no solution?

Look for situations where you need a positive quantity to be negative. Examples: $x^2 < -5$ , $|x| < -3$ , or sums of absolute values less than negative numbers.

Should I still try to solve it step by step?

No need! Once you recognize that $|a| + |b| \ge 0$ but the inequality requires < -1, you can immediately conclude no solution exists.

What's the difference between 'no solution' and 'all real numbers'?

No solution: The inequality is impossible (like our problem). All real numbers: Every value works (like $|x| \ge 0$ ).

🌟 Unlock Your Math Potential

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