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Chapter 5: Problem 112

Simplify. \(\frac{1}{2} x-\frac{1}{4}-\left(\frac{3}{2} x+\frac{3}{4}\right)\)

Short Answer

Expert verified
-x - 1

Step by step solution

01

- Distribute the negative sign

The given expression is \(\frac{1}{2} x - \frac{1}{4} - \big(\frac{3}{2} x + \frac{3}{4}\big)\). First, distribute the negative sign inside the parentheses: \(\frac{1}{2} x - \frac{1}{4} - \frac{3}{2} x - \frac{3}{4}\).
02

- Combine like terms

Next, combine the like terms in the expression. For the x terms: \(\frac{1}{2} x - \frac{3}{2} x = -\frac{2}{2} x = -x\). For the constant terms: \(-\frac{1}{4} - \frac{3}{4} = -\frac{4}{4} = -1\).
03

- Simplify

Now, write the simplified expression by combining the results of the like terms: \(-x - 1\).

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Key Concepts

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

distributing negative signs
When you see a negative sign in front of a parenthesis, it means you need to distribute that negative sign to each term inside the parenthesis. In our exercise, we have the expression \(\frac{1}{2} x - \frac{1}{4} - \big(\frac{3}{2} x + \frac{3}{4}\big)\).
To distribute the negative sign, you need to multiply each term within the parenthesis by -1. Hence, the expression becomes \(\frac{1}{2} x - \frac{1}{4} - \frac{3}{2} x - \frac{3}{4}\).
This step is crucial because it changes the signs of the terms within the parenthesis. Remember, whenever you distribute a negative sign, always change the sign of each term it affects.
combining like terms
Next, we need to simplify our expression by combining like terms. Like terms are terms that have the same variable raised to the same power, or they are constants.
In our expression \(\frac{1}{2} x - \frac{1}{4} - \frac{3}{2} x - \frac{3}{4}\), we have two types of like terms: the x terms and the constant terms.
Let's start with the x terms: \(\frac{1}{2} x - \frac{3}{2} x\).
Subtracting these gives us: \- \frac{2}{2} x = -x \.
Now, combine the constant terms: \(-\frac{1}{4} - \frac{3}{4} = -\frac{4}{4} = -1\).
This step helps to gather all similar elements together, making the expression simpler to handle.
simplification process
The final step involves writing the simplified form of the given expression after distributing the negative sign and combining like terms.
From the previous steps, we have: \(-x - 1\).
So, the simplified form of the expression \(\frac{1}{2} x - \frac{1}{4} - \big(\frac{3}{2} x + \frac{3}{4}\big)\) is \(-x - 1\).
The simplification process ensures that the expression is as concise as possible, making it easier to understand and work with.

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Most popular questions from this chapter

Computer spreadsheet applications allow values for cells in a spreadsheet to be calculated from values in other cells. For example, if the cell Cl contains the formula $$ =\mathrm{A} 1+2 * \mathrm{B} 1 $$ the value in Cl will be the sum of the value in Al and twice the value in B1. This formula is a polynomial in the two variables Al and B1. The cell D6 contains the formula $$ =\mathrm{Al}-0.2 * \mathrm{B} 1+0.3^{*} \mathrm{Cl} $$ What is the value in \(\mathrm{D} 6\) if the value in \(\mathrm{Al}\) is \(10,\) the value in \(\mathrm{B} 1\) is \(-3,\) and the value in \(\mathrm{Cl}\) is \(30 ?\)

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=\frac{1}{2-x}\\\ &g(x)=7-x \end{aligned} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=x^{3}+1\\\ &g(x)=\frac{5}{x} \end{aligned} $$

To prepare for Section \(5.2,\) review operations with integers (Sections \(1.5-1.7)\) Perform the indicated operations. $$ -3+(-11) $$

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=5+x\\\ &g(x)=6-2 x \end{aligned} $$
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