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Chapter 1: Problem 10

If $$\frac{2}{5}(5 x)+2(x-1)=4(x+1)-2,$$ what is the value of \(x ?\) $$ \begin{array}{l}{\text { (A) } x=-2} \\ {\text { (B) } x=2} \\ {\text { (C) There is no value of } x \text { for which the equation is }} \\ {\text { true. }} \\ {\text { (D) There are infinitely many values of } x \text { for which }} \\ {\text { the equation is true. }}\end{array} $$

Short Answer

Expert verified
C. There is no value of \( x \) for which the equation is true.

Step by step solution

01

- Simplify the equation

The given equation is \( \frac{2}{5}(5 x) + 2(x - 1) = 4(x + 1) - 2 \). First, distribute \( \frac{2}{5} \times 5x \), \( 2(x-1) \) and \( 4(x+1) \). This simplifies to: \( 2x + 2x - 2 = 4x + 4 - 2 \)
02

- Combine like terms

Combine the like terms on each side of the equation. The left side simplifies to \( 2x + 2x - 2 \to 4x - 2 \). The right side simplifies to \( 4x + 4 - 2 \to 4x + 2 \). So, the equation simplifies to: \( 4x - 2 = 4x + 2 \).
03

- Subtract \(4x\) from both sides

Subtract \(4x\) from both sides to simplify further: \( 4x - 2 - 4x = 4x + 2 - 4x \). This simplifies the equation to: \( -2 = 2 \).
04

- Analyze the result

The simplified equation \( -2 = 2 \) is a contradiction, which means there is no value of \( x \) that can satisfy the original equation.

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Key Concepts

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

Contradiction in Equations
A contradiction in equations occurs when, after simplifying and solving an equation, we arrive at a statement that is always false.
For example, in the given exercise, we end up with the equation \(-2 = 2\).
This is clearly a contradiction because no value of x will ever make this statement true.
When a contradiction occurs, it means that there is no solution to the equation.
It indicates that the equation has no values of x that satisfy it. Such equations are often described as having 'no solution.'
Combining Like Terms
Combining like terms is a crucial step in simplifying equations.
Like terms are terms that have the same variable raised to the same power.
In the exercise, we see this when we combine \(2x\) and \(2x\) on the left side to get \(4x\).
Similarly, \(4x\) and \4 on the right side remain as is. The goal is to make the equation simpler.
By adding or subtracting these like terms, we can condense the equation, making it easier to solve.
Combining like terms is essential for keeping the equation balanced and in a more straightforward form.
Distributive Property
The distributive property allows us to remove parentheses in an equation by distributing a multiplication over addition or subtraction inside the parentheses.
The general form is a(b + c) = ab + ac.
In the given exercise, we use the distributive property multiple times: \( \frac{2}{5}(5x) + 2(x - 1) = 4(x + 1) - 2 \).
Distributing \(\frac{2}{5} \) over \(5x \) gives us \2x\. Distributing \2 \ over \(x - 1 \) results in \( 2x - 2 \).
Similarly, distributing \4 \ over \(x + 1 \) gives us \( 4x + 4 \).
The distributive property simplifies complex equations, allowing us to combine like terms and solve for the variable.
No Solution Condition
An equation is said to have no solution when it results in a statement that is always false, like \( -2 = 2 \).
This means that no matter what value we substitute for the variable, the equation will never be true.
To identify a no solution condition:
  • Simplify the equation as much as possible.
  • Combine like terms and use the distributive property where necessary.
  • If you reach a false statement after all steps, the equation has no solution.
The given exercise reveals a no solution condition because after simplifying, we end up with \( -2 = 2 \), a clear contradiction.
Thus, it confirms that no value of \x\ will satisfy the given equation.

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Most popular questions from this chapter

$$\frac{3}{4} x-\frac{1}{2} y=12$$ $$k x-2 y=22$$ If the system of linear equations above has no solution, and \(k\) is a constant, what is the value of \(k ?\) $$\begin{array}{l}{\text { (A) }-\frac{4}{3}} \\ {\text { (B) }-\frac{3}{4}} \\\ {\text { (C) } 3} \\ {\text { (D) } 4}\end{array}$$

The percent increase from 5 to 12 is equal to th percent increase from 12 to what number? $$ \begin{array}{l}{\text { (A) } 16.8} \\ {\text { (B) } 19.0} \\ {\text { (C) } 26.6} \\ {\text { (D) } 28.8}\end{array} $$

In Delray Beach, Florida, you can take a luxury golf cart ride around downtown. The driver charges \(\$ 4\) for the first \(\frac{1}{4}\) mile, plus \(\$ 1.50\) for each additional \(\frac{1}{2}\) mile. Which inequality represents the number of miles, \(m,\) that you could ride and pay no more than \(\$ 10 ?\) $$\begin{array}{l}{\text { (A) } 3.25+1.5 m \leq 10} \\ {\text { (B) } 3.25+3 m \leq 10} \\ {\text { (C) } 4+1.5 m \leq 10} \\ {\text { (D) } 4+3 m \leq 10}\end{array}$$

$$\frac{3.86}{x}+\frac{180.2}{10 x}+\frac{42.2}{5 x}$$ The Ironman Triathlon originated in Hawaii in \(1978 .\) The format of the Ironman has not changed since then: It consists of a \(3.86-\mathrm{km}\) swim, a \(180.2-\mathrm{km}\) bicycle ride, and a \(42.2-\mathrm{km}\) run, all raced in that order and without a break. Suppose an athlete bikes 10 times as fast as he swims and runs 5 times as fast as he swims. The variable \(x\) in the expression above represents the rate at which the athlete swims, and the whole expression represents the number of hours that it takes him to complete the race. If it takes him 16.2 hours to complete the race, how many kilometers did he swim in 1 hour? $$\begin{array}{l}{\text { (A) } 0.85} \\ {\text { (B) } 1.01} \\ {\text { (C) } 1.17} \\ {\text { (D) } 1.87}\end{array}$$

$$\frac{1}{x}+\frac{4}{x}=\frac{1}{72}$$ In order to create safe drinking water, cities and towns use water treatment facilities to remove contaminants from surface water and groundwater. Suppose a town has a treatment plant but decides to build a second, more efficient facility. The new treatment plant can filter the water in the reservoir four times as quickly as the older facility. Working together, the two facilities can filter all the water in the reservoir in 72 hours. The equation above represents the scenario. Which of the following describes what the term \(\frac{1}{x}\) represents? (A) The portion of the water the older treatment plant can filter in 1 hour (B) The time it takes the older treatment plant to filter the water in the reservoir (C) The time it takes the older treatment plant to filter \(\frac{1}{72}\) of the water in the reservoir (D) The portion of the water the new treatment plant can filter in 4 hours
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