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Chapter 0: Problem 44

In Exercises 39-48, evaluate the expression. $$ -3-|-3| $$

Short Answer

Expert verified
-6

Step by step solution

01

Evaluate Absolute Value

Evaluate the absolute value of -3, denoted by |-3|. The absolute value of any number is the non-negative value of the number. For our expression, |-3| equals to 3.
02

Subtract the Result

Now, substitute |-3| with 3 in the original expression. So, we have -3 - 3. Perform the subtraction operation to obtain the result.

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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 represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, like \(|x|\). The absolute value is always non-negative. For example:
  • The absolute value of \(-3\) is \(|-3| = 3\).
  • Similarly, the absolute value of \(+3\) is \(|3| = 3\).

When you see an expression such as \(|-x|\), convert it to a positive, thus dismissing any negative sign. This helps in simplifying expressions that involve absolute values. It's essential to recognize that absolute values can change the computation outcomes, especially when dealing with negative numbers.
The Subtraction Process
Subtraction is one of the basic arithmetic operations where we find the difference between numbers by deducting one from another. In the expression \(-3 - |3|\), subtraction applies after evaluating the absolute value, following the order of operations (PEMDAS/BODMAS).
  • The expression becomes \(-3 - 3\), after substituting the absolute value.
  • Perform the subtraction: Finally, compute \(-3 - 3\), which gives \(-6\).

It's critical to maintain the sign of the numbers while performing subtraction. This step-by-step approach helps in accurately solving expressions involving subtraction, especially when mixed with other operations.
The Role of Negative Numbers
Negative numbers play a pivotal role in mathematics, extending the number system to the left side of the number line. They require careful handling, particularly with operations like subtraction.
  • When you subtract a positive number from a negative number, the result becomes more negative. For example, \(-3 - 3 = -6\).
  • Understanding their position on the number line: Negative numbers are all values less than zero, such as \(-1, -2, -3, \ldots\).

Working with negative numbers involves recognizing that subtracting a positive value results in moving further left on the number line. Therefore, correctly managing these steps ensures the accurate evaluation of expressions including negative numbers.

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Most popular questions from this chapter

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=-\sqrt{x+1}+3 &\quad x_{1}=3, x_{2}=8 \end{array} $$

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from a height of 6 feet at a velocity of 64 feet per second. $$ t_{1}=0, t_{2}=3 $$

(a) The amount in your savings account is a function of your salary. (b) The speed at which a free-falling baseball strikes the ground is a function of the height from which it was dropped.

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{3 / 5} $$

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\sqrt{2 x+3} $$
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