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Chapter 5: Problem 66

Simplify the given expression. \(15 .88-|-5.22-(-19.94)|\)

Short Answer

Expert verified
1.16

Step by step solution

01

Solve Inside the Absolute Value

First, address the expression inside the absolute value brackets: \[-5.22 - (-19.94)\] Subtracting a negative is equivalent to adding its positive. Thus, \[-5.22 + 19.94 = 14.72\].
02

Apply Absolute Value

Now, apply the absolute value which turns any expression inside it into a positive value:\[|-14.72| = 14.72\].In this case, the expression is already positive, so the absolute value function doesn鈥檛 change the result.
03

Perform the Subtraction

Subtract the result from Step 2 from the first term in the expression:\[15.88 - 14.72\] Carefully subtract to find the simplified result:\[15.88 - 14.72 = 1.16\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Absolute Value
Absolute value is a fundamental mathematical concept that transforms any number into its non-negative form. When you see absolute value notation, it appears with vertical bars around the number or expression inside, like this: \(|x|\). The absolute value of a number is essentially its distance from zero on the number line, ignoring any negative sign.
  • For positive numbers and zero: \(|x| = x\)
  • For negative numbers: \(|x| = -x\)
In the exercise, we encounter the expression \(|-14.72|\). Here, because absolute value makes any number positive, it results in 14.72. This step is crucial since it affects the subsequent calculations.
Fundamentals of Addition and Subtraction
Addition and subtraction are basic arithmetic operations vital for solving expressions. When dealing with addition and subtraction, we must carefully follow the signs (positive or negative) of the numbers involved.
  • Adding a negative is akin to subtraction.
  • Subtracting a negative turns into addition.
In the context of the problem, the expression \(-5.22 - (-19.94)\) is simplified by changing the subtraction of a negative to addition: \(-5.22 + 19.94\). This is a common scenario where knowing how to handle negative signs simplifies the process. It turns what appears to be a complex subtraction into straightforward addition, which gives us 14.72.
Simplifying Arithmetic Expressions
Simplifying expressions involves combining like terms and resolving any operations step by step to reach the simplest form. In this exercise, you have already calculated the expression inside the absolute value. The next step is continuing with the remaining arithmetic operations outside the absolute value.
  • Always follow the order of operations (PEMDAS/BODMAS).
  • Ensure to perform addition and subtraction after dealing with any brackets, exponents, or multiplication/division when they are present.
The final step in the example provided involves taking \(15.88 - 14.72\). Doing this subtraction leads us to a simple result: 1.16. By carefully addressing each component within expressions, we simplify them to their most basic form. This process eliminates complexity and helps us arrive at a clear answer.

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