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Chapter 0: Problem 1

Evaluate \(|-4|+|4|\).

Short Answer

Expert verified
To evaluate the expression \(|-4| + |4|\), first find the absolute values of -4 and 4, which are both 4. Then add the absolute values: \(4 + 4 = 8\). The final answer is \(8\).

Step by step solution

01

Evaluate Absolute Values

Firstly, we will find the absolute values of -4 and 4. Recall that the absolute value of a number is its distance from 0 without considering the sign. In mathematical terms, the absolute value of a number x is represented as \(|x|\). So, we have \(|-4|\) and \(|4|\). Since both -4 and 4 are 4 units away from 0, their absolute values are equal to 4. Therefore, \(|-4| = 4\) and \(|4| = 4\).
02

Add the Absolute Values

Now that we have found the absolute values of -4 and 4, we will add them together: \(|-4| + |4| = 4 + 4\)
03

Calculate the Sum

Finally, we will calculate the sum of the absolute values: \(4 + 4 = 8\) So, the final answer for the expression \(|-4|+|4|\) is \(8\).

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Key Concepts

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

Distance from Zero
The absolute value of a number represents its distance from zero on the number line. Whether the number is positive or negative, its absolute value is always non-negative. For example:
  • The absolute value of \(-4\) is \(|-4| = 4\), as it is 4 units away from zero.
  • Similarly, the absolute value of \(4\) is also \(|4| = 4\).

The concept of absolute value is like asking "how far is this number from zero?" without worrying about direction. It strips away any negative signs, giving you a straightforward measure of distance. Understanding this is crucial when working with absolute values, as it helps simplify many mathematical problems.
Addition of Absolute Values
Once absolute values are evaluated, as in our previous step, they can be added just like any regular numbers. In our example:
  • We determined \(|-4| = 4\) and \(|4| = 4\).
  • These were added together: \(|-4| + |4| = 4 + 4\).

When dealing with absolute values in addition, remember:
  • Treat each absolute value as a separate positive number, regardless of the original sign of the number inside the absolute value brackets.
  • Simply add these evaluated absolute values together.

This is an integral part of many math problems, helping break down complex expressions into manageable parts.
Evaluating Expressions
Evaluating expressions involves performing operations in a specific order to find the expression's value. With absolute values, as shown in the exercise, the first step is always to evaluate the absolute values themselves:
  • Determine the absolute values, turning negative signs into positive numbers.
  • Proceed with arithmetic operations like addition or subtraction as shown: \(4 + 4 = 8\).

By following a structured approach:
  • The process begins with resolving absolute values.
  • Next, perform any additions, subtractions, or other operations.

This clear sequence aids in solving the expression smoothly and accurately, ultimately ensuring a correct and precise solution.

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Most popular questions from this chapter

Suppose \(b \neq 0\) and \(d \neq 0\). Explain why \(\frac{a}{b}=\frac{c}{d}\) if and only if \(a d=b c .\)

Write each set as an interval or as a union of two intervals. $$\\{x:|x|>2\\}$$

Explain why $$|a b|=|a||b|$$ for all real numbers \(a\) and \(b\).

Give four examples of pairs of real numbers \(a\) and \(b\) such that \(|a+b|=2\) and \(|a|+|b|=8\).

Explain why $$|-a|=|a|$$ for all real numbers \(a\).
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