Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
Solve algebraically for x
![### Solving Multi-Step Equations with Fractions
#### Problem:
\[ -\frac{2}{3}(x + 12) + \frac{2}{3}x = -\frac{5}{4}x + 2 \]
In this equation, we have a combination of fractions and variables that we need to solve for \( x \). Follow these steps to simplify and solve the equation.
#### Steps to Solve:
1. **Distribute the fractions:**
Apply the distributive property to \(-\frac{2}{3}(x + 12)\):
\[
-\frac{2}{3}x - \frac{2}{3} \cdot 12 + \frac{2}{3}x = -\frac{5}{4}x + 2
\]
Simplify the product:
\[
-\frac{2}{3}x - 8 + \frac{2}{3}x = -\frac{5}{4}x + 2
\]
2. **Combine like terms:**
Notice that \(-\frac{2}{3}x + \frac{2}{3}x\) on the left side cancels out:
\[
-8 = -\frac{5}{4}x + 2
\]
3. **Isolate the variable term:**
Subtract 2 from both sides to get the variable term by itself:
\[
-8 - 2 = -\frac{5}{4}x
\]
Simplify:
\[
-10 = -\frac{5}{4}x
\]
4. **Solve for \( x \):**
Multiply both sides by the reciprocal of \(-\frac{5}{4}\), which is \(-\frac{4}{5}\):
\[
-10 \times -\frac{4}{5} = x
\]
Solve:
\[
x = 8
\]
Thus, the solution to the equation is:
\[ x = 8 \]
#### Explanation:
- We first used the distributive property to eliminate the parentheses.
- Combined like terms to simplify the equation further.
- Isolated the variable \( x \) on one side of the equation.
- Finally, solved for \( x \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ff6b2c7-34e8-46c3-909d-02571dbb7c3e%2Ffce1ba59-353b-4317-8a2a-2749d4f4c7ca%2Fyv6w8w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Multi-Step Equations with Fractions
#### Problem:
\[ -\frac{2}{3}(x + 12) + \frac{2}{3}x = -\frac{5}{4}x + 2 \]
In this equation, we have a combination of fractions and variables that we need to solve for \( x \). Follow these steps to simplify and solve the equation.
#### Steps to Solve:
1. **Distribute the fractions:**
Apply the distributive property to \(-\frac{2}{3}(x + 12)\):
\[
-\frac{2}{3}x - \frac{2}{3} \cdot 12 + \frac{2}{3}x = -\frac{5}{4}x + 2
\]
Simplify the product:
\[
-\frac{2}{3}x - 8 + \frac{2}{3}x = -\frac{5}{4}x + 2
\]
2. **Combine like terms:**
Notice that \(-\frac{2}{3}x + \frac{2}{3}x\) on the left side cancels out:
\[
-8 = -\frac{5}{4}x + 2
\]
3. **Isolate the variable term:**
Subtract 2 from both sides to get the variable term by itself:
\[
-8 - 2 = -\frac{5}{4}x
\]
Simplify:
\[
-10 = -\frac{5}{4}x
\]
4. **Solve for \( x \):**
Multiply both sides by the reciprocal of \(-\frac{5}{4}\), which is \(-\frac{4}{5}\):
\[
-10 \times -\frac{4}{5} = x
\]
Solve:
\[
x = 8
\]
Thus, the solution to the equation is:
\[ x = 8 \]
#### Explanation:
- We first used the distributive property to eliminate the parentheses.
- Combined like terms to simplify the equation further.
- Isolated the variable \( x \) on one side of the equation.
- Finally, solved for \( x \)
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