Understanding Absolute Value Equality: Checking |16 - 5| = |8 + 3| = 11

When working with absolute values in mathematics, one essential skill is verifying whether two expressions inside the absolute value symbols are truly equal. In this article, we explore and validate the equation:

LHS = |16 - 5| = 11
RHS = |8 + 3| = 11
Is this statement valid? Let’s break it down step by step.

Understanding the Context

What is Absolute Value?

The absolute value of a number represents its distance from zero on the number line, regardless of direction. This means:

  • |a| = a if a ≥ 0
  • |a| = -a if a < 0
Check: LHS = |16 - 5| = 11, RHS = |8 + 3| = 11 → valid.

Key Insights

In both cases, the result is always non-negative.

Step 1: Simplify the Left-Hand Side (LHS)

LHS = |16 - 5|
First, compute the difference inside the absolute value:
16 - 5 = 11

Now apply absolute value:
|11| = 11
So, LHS = 11

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A project manager is overseeing two teams working on a new software feature. Team A works 8 hours a day for 5 days and completes 40% of the project. Team B works 7 hours a day for 6 days and completes 30% of the project. If both teams continue at the same rate, how many total hours will it take for both teams combined to complete the entire project?
Team A completes 40% in \(8 \times 5 = 40\) hours. So, 40% = 40 hours → 1% = 1 hour → 100% = 100 hours.

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Final Thoughts

Step 2: Simplify the Right-Hand Side (RHS)

RHS = |8 + 3|
First, compute the sum inside the absolute value:
8 + 3 = 11

Now apply absolute value:
|11| = 11
So, RHS = 11

Step 3: Compare Both Sides

We have:
LHS = 11 and RHS = 11
Since both sides are equal, the equation |16 - 5| = |8 + 3| holds true.

Why This Matters: Checking Validity with Absolute Values

Absolute value expressions often appear in geometry, algebra, and equations involving distances. Validating such expressions ensures accuracy in problem-solving and protects against errors in calculations. The key takeaway is:
If |A| = |B|, then A = B or A = -B. But in this case, LHS = RHS = 11, confirming equality directly.