Evaluate $|a| - |b|$, Given $a = 5$, $b = -3$, And $c = -2$.A. 8 B. 2 C. 7 D. 3

Introduction

In mathematics, absolute value is a fundamental concept that deals with the distance of a number from zero on the number line. It is denoted by two vertical lines, and its value is always non-negative. In this article, we will evaluate the expression $|a| - |b|$, given $a = 5$, $b = -3$, and $c = -2$. We will break down the problem step by step and provide a clear explanation of the solution.

Understanding Absolute Value

Before we dive into the problem, let's recall the definition of absolute value. The absolute value of a number $x$, denoted by $|x|$, is defined as:

$$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$

In other words, if the number is positive or zero, its absolute value is the number itself. If the number is negative, its absolute value is the negative of the number.

Evaluating the Expression

Now that we have a clear understanding of absolute value, let's evaluate the expression $|a| - |b|$. We are given that $a = 5$ and $b = -3$. To evaluate the expression, we need to find the absolute values of $a$ and $b$.

Step 1: Evaluate the Absolute Value of $a$

Since $a = 5$ is a positive number, its absolute value is simply $a$ itself.

$$|a| = |5| = 5$$

Step 2: Evaluate the Absolute Value of $b$

Since $b = -3$ is a negative number, its absolute value is the negative of the number.

$$|b| = |-3| = -(-3) = 3$$

Step 3: Evaluate the Expression $|a| - |b|$

Now that we have found the absolute values of $a$ and $b$, we can evaluate the expression $|a| - |b|$.

$$|a| - |b| = 5 - 3 = 2$$

Conclusion

In this article, we evaluated the expression $|a| - |b|$, given $a = 5$, $b = -3$, and $c = -2$. We broke down the problem step by step and provided a clear explanation of the solution. We found that the absolute value of $a$ is $5$, the absolute value of $b$ is $3$, and the expression $|a| - |b|$ evaluates to $2$.

Final Answer

The final answer is $\boxed{2}$.

Additional Examples

To reinforce your understanding of absolute value, let's consider a few more examples.

• Evaluate the expression $|x| - |y|$, given $x = 2$ and $y = -4$.
• Evaluate the expression $|z| - |w|$, given $z = -1$ and $w = 3$.

Solutions

• $|x| - |y| = |2| - |-4| = 2 - 4 = -2$
• $|z| - |w| = |-1| - |3| = 1 - 3 = -2$

Conclusion

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Introduction

In our previous article, we evaluated the expression $|a| - |b|$, given $a = 5$, $b = -3$, and $c = -2$. We broke down the problem step by step and provided a clear explanation of the solution. In this article, we will answer some frequently asked questions about absolute value expressions.

Q&A

Q: What is the absolute value of a negative number?

A: The absolute value of a negative number is the positive of the number. For example, the absolute value of $-3$ is $3$.

Q: How do I evaluate an absolute value expression with two variables?

A: To evaluate an absolute value expression with two variables, you need to find the absolute values of each variable separately. Then, you can substitute the absolute values into the expression and simplify.

Q: What is the difference between the absolute value of a number and the number itself?

A: The absolute value of a number is always non-negative, while the number itself can be positive, negative, or zero.

Q: Can I use absolute value to solve equations with variables?

A: Yes, you can use absolute value to solve equations with variables. However, you need to consider two cases: one where the variable is positive and one where the variable is negative.

Q: How do I evaluate an absolute value expression with a variable and a constant?

A: To evaluate an absolute value expression with a variable and a constant, you need to find the absolute value of the variable and then substitute it into the expression.

Q: What is the absolute value of zero?

A: The absolute value of zero is zero.

Q: Can I use absolute value to solve inequalities with variables?

A: Yes, you can use absolute value to solve inequalities with variables. However, you need to consider two cases: one where the variable is positive and one where the variable is negative.

Q: How do I evaluate an absolute value expression with multiple variables?

A: To evaluate an absolute value expression with multiple variables, you need to find the absolute values of each variable separately and then substitute them into the expression.

Examples

Example 1: Evaluating an Absolute Value Expression with Two Variables

Evaluate the expression $|x| - |y|$, given $x = 2$ and $y = -4$.

Solution:

• Find the absolute values of $x$ and $y$: $|x| = |2| = 2$ and $|y| = |-4| = 4$
• Substitute the absolute values into the expression: $|x| - |y| = 2 - 4 = -2$

Example 2: Evaluating an Absolute Value Expression with a Variable and a Constant

Evaluate the expression $|z| - 3$, given $z = -1$.

Solution:

• Find the absolute value of $z$: $|z| = |-1| = 1$
• Substitute the absolute value into the expression: $|z| - 3 = 1 - 3 = -2$

Example 3: Evaluating an Absolute Value Expression with Multiple Variables

Evaluate the expression $|x| - |y| + |z|$, given $x = 2$, $y = -4$, and $z = -1$.

Solution:

• Find the absolute values of $x$, $y$, and $z$: $|x| = |2| = 2$, $|y| = |-4| = 4$, and $|z| = |-1| = 1$
• Substitute the absolute values into the expression: $|x| - |y| + |z| = 2 - 4 + 1 = -1$

Conclusion

In this article, we answered some frequently asked questions about absolute value expressions. We provided examples to illustrate how to evaluate absolute value expressions with two variables, a variable and a constant, and multiple variables. We also discussed how to use absolute value to solve equations and inequalities with variables.