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Chapter 6: Problem 44

Solve each equation, and check the solutions. $$ \frac{4}{x}+\frac{1}{x}=2 $$

Short Answer

Expert verified
x = \( \frac{5}{2} \)

Step by step solution

01

Combine the Fractions

Combine the fractions on the left side of the equation: \[ \frac{4}{x} + \frac{1}{x} = \frac{4 + 1}{x} = \frac{5}{x} \] The equation now becomes: \[ \frac{5}{x} = 2 \]
02

Solve for x

To solve for \( x \), multiply both sides by \( x \) to isolate the variable: \[ 5 = 2x \]
03

Isolate x

Divide both sides by 2 to solve for \( x \): \[ x = \frac{5}{2} \]
04

Check the Solution

Substitute \( x = \frac{5}{2} \) back into the original equation to verify: \[ \frac{4}{\frac{5}{2}} + \frac{1}{\frac{5}{2}} = 2 \] Simplify the fractions: \[ \frac{8}{5} + \frac{2}{5} = 2 \] \[ \frac{8 + 2}{5} = 2 \] \[ \frac{10}{5} = 2 \] \[ 2 = 2 \] The solution checks out.

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Key Concepts

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

Combining Fractions
To solve rational equations, you often need to combine fractions. If the fractions have the same denominator, this process becomes quite simple. You only sum the numerators and place the result over the common denominator. In the initial problem \( \frac{4}{x} + \frac{1}{x} = 2 \), both fractions already have the denominator \( x \). So, we can directly add the numerators (4 and 1). This gives us:
\[ \frac{4+1}{x} = \frac{5}{x} \]
Now we have a simpler expression to work with:
\[ \frac{5}{x} = 2 \]
Multiplying Both Sides of an Equation
When you have a fraction in an equation, one effective way to isolate the variable is to multiply both sides of the equation by the denominator of the fraction. For the simplified equation \( \frac{5}{x} = 2 \), the denominator is \( x \). Thus, multiplying both sides by \( x \):
\[ x \times \frac{5}{x} = x \times 2 \]
This will cancel out \( x \) on the left side, leaving you with:
\[ 5 = 2x \]
Multiplying by the denominator is a powerful tool in solving rational equations as it transforms the fraction into a more straightforward algebraic expression.
Isolating the Variable
Once you have an equation without fractions, the next goal is to isolate the variable (usually denoted as \( x \)). In our case, after multiplying both sides of \( \frac{5}{x} = 2 \) by \( x \), we got \[ 5 = 2x \]. To isolate \[ x \], divide both sides of the equation by 2:
\[ \frac{5}{2} = x \]
This step ensures we separate \[ x \] on one side of the equation, making it clear what \[ x \] equals.
Checking Solutions
After solving for \ x \, it is crucial to verify your solution. Start by substituting \ x \ back into the original equation. For \ x = \frac{5}{2} \, check by plugging it into:
\[ \frac{4}{\frac{5}{2}} + \frac{1}{\frac{5}{2}} = 2 \]
Simplifying, we get:
\[ \frac{8}{5} + \frac{2}{5} = 2 \]
Summing the fractions results in:
\[ \frac{10}{5} = 2 \]
which simplifies to \[ 2 = 2 \]. This confirms the solution is correct. Always check your solution to ensure there are no mistakes in your calculations.

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Most popular questions from this chapter

Solve each formula or equation for the specified variable. $$ \frac{5}{p}+\frac{2}{q}+\frac{3}{r}=1 \text { for } r $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{1}{m^{3} p}+\frac{2}{m p^{2}}}{\frac{4}{m p}+\frac{1}{m^{2} p}} $$

Simplify each fraction. $$ \frac{1+t^{-1}-56 t^{-2}}{1-t^{-1}-72 t^{-2}} $$

Solve each formula or equation for the specified variable. $$ \frac{t}{x-1}-\frac{2}{x+1}=\frac{1}{x^{2}-1} \text { for } t $$

Let \(P\), \(Q\), and \(R\) be rational expressions defined as follows. $$P=\frac{6}{x+3}, \quad Q=\frac{5}{x+1}, \quad R=\frac{4 x}{x^{2}+4 x+3}$$ Why is \((P \cdot Q) \div R\) not defined if \(x=0 ?\)
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