Solve |a|-|18-9|+|4|<0: Multiple Absolute Value Inequality Analysis

Q: Why do I need to evaluate the constant absolute values first?

Evaluating constants like $ |18-9| = 9 $ and $ |4| = 4 $ simplifies your work! This turns the complex expression into \( |a| - 5 , which is much easier to solve.

Q: How do I solve |a| < 5?

When \( |a| , it means a is less than 5 units away from zero. This gives us \( -5 , so a can be any number between -5 and 5.

Q: What if I got |a| > 5 instead?

If you have $ |a| > 5 $, then a would be more than 5 units away from zero. This means $ a or \( a > 5 $.

Q: How can I check my answer?

Test values from your solution! Try $ a = 0 $: $ |0| - 9 + 4 = -5 ✓. Try \( a = 6 $: $ |6| - 9 + 4 = 1 > 0 $ ✗

Q: Why isn't a = 5 included in the solution?

Because we need strict inequality ($ a = 5 $, we get $ |5| - 5 = 0 $, but we need the result to be less than 0, not equal to 0.

Absolute Value Inequalities with Multiple Terms

Given:

$|a|-|18-9|+|4|<0$

Which of the following statements is necessarily true?

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Step-by-step video solution

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Step-by-step written solution

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Understand the problem

Given:

$|a|-|18-9|+|4|<0$

Which of the following statements is necessarily true?

Step-by-step solution

To solve this problem, we'll follow these steps:

Simplify the expression inside the inequality.
Determine the necessary condition for $a$ .
Compare the result with the provided options.

Step 1: Simplify the expression.
We start by evaluating the fixed absolute values:
$|18 - 9| = 9$ and $|4| = 4$ .
Substituting these values into the inequality gives us:
$|a| - 9 + 4 < 0.$

Step 2: Simplify further and solve for $|a|$ .
Combine constants:
$|a| - 5 < 0$
Thus, we have:
$|a| < 5$ .

Step 3: Apply the property of absolute values.
The inequality $|a| < 5$ implies that:
$-5 < a < 5$ .

Therefore, the solution to the problem is $-5 < a < 5$ .

Final Answer

$-5 < a < 5$

Key Points to Remember

Essential concepts to master this topic

Rule: Evaluate absolute values inside expressions before solving inequalities
Technique: Simplify |18-9| = 9 and |4| = 4 first
Check: Test boundary values: when a = 5, |5| - 5 = 0 (not < 0) ✓

Common Mistakes

Avoid these frequent errors

Forgetting to simplify constant absolute values first
Don't try to solve |a| - |18-9| + |4| < 0 without evaluating the constants first = confusing expressions! This makes the problem unnecessarily complex. Always simplify |18-9| = 9 and |4| = 4 before working with the variable.

Practice Quiz

Test your knowledge with interactive questions

Given:

$\left|2x-1\right|>-10$

Which of the following statements is necessarily true?

FAQ

Everything you need to know about this question

Why do I need to evaluate the constant absolute values first?

Evaluating constants like $|18-9| = 9$ and $|4| = 4$ simplifies your work! This turns the complex expression into $|a| - 5 < 0$ , which is much easier to solve.

How do I solve |a| < 5?

When $|a| < 5$ , it means a is less than 5 units away from zero. This gives us $-5 < a < 5$ , so a can be any number between -5 and 5.

What if I got |a| > 5 instead?

If you have $|a| > 5$ , then a would be more than 5 units away from zero. This means $a < -5$ or $a > 5$ .

How can I check my answer?

Test values from your solution! Try $a = 0$ : $|0| - 9 + 4 = -5 < 0$ ✓. Try $a = 6$ : $|6| - 9 + 4 = 1 > 0$ ✗

Why isn't a = 5 included in the solution?

Because we need strict inequality (<). When $a = 5$ , we get $|5| - 5 = 0$ , but we need the result to be less than 0, not equal to 0.

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Solve |a|-|18-9|+|4|<0: Multiple Absolute Value Inequality Analysis

Absolute Value Inequalities with Multiple Terms

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to evaluate the constant absolute values first?

How do I solve |a| < 5?

What if I got |a| > 5 instead?

How can I check my answer?

Why isn't a = 5 included in the solution?

🌟 Unlock Your Math Potential

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