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Chapter 6: Problem 31

Solve each equation, and check your solutions. \(\frac{2 x+3}{x}=\frac{3}{2}\)

Short Answer

Expert verified
The solution is \(x = -6\).

Step by step solution

01

Set up the Equation

Start with the given equation: \[\frac{2x + 3}{x} = \frac{3}{2}.\]
02

Cross-Multiply

To eliminate the fractions, cross-multiply: \[2(2x + 3) = 3x.\]
03

Distribute and Simplify

Distribute the 2 on the left side and then simplify: \[4x + 6 = 3x.\]
04

Solve for x

Subtract 3x from both sides to isolate x: \[x + 6 = 0.\]Next, subtract 6 from both sides: \[x = -6.\]
05

Check the Solution

Substitute \(x = -6\) back into the original equation to verify: \[\frac{2(-6) + 3}{-6} = \frac{3}{2}\]Simplify the left side: \[\frac{-12 + 3}{-6} = \frac{-9}{-6} = \frac{3}{2}.\]Since both sides of the equation are equal, \(x = -6\) is the correct solution.

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Key Concepts

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

cross-multiplication in algebra
Cross-multiplication is a powerful technique used to solve rational equations. When you have an equation of the form \(\frac{A}{B} = \frac{C}{D}\), you can eliminate the fractions by 'cross-multiplying.' Simply multiply the numerator of one fraction by the denominator of the other, and vice versa.
For example, in the equation \(\frac{2x + 3}{x} = \frac{3}{2}\), you cross-multiply as follows: \[2 \times (2x + 3) = 3 \times x\].
This gives you an equation without any denominators, making it easier to solve.
isolating variables
Isolating the variable is a crucial step in solving any equation. Once we have cross-multiplied, the equation becomes \[4x + 6 = 3x\].
The goal is to get the variable (in this case, x) by itself on one side of the equation. Start by eliminating like terms on both sides. Here, we subtract 3x from both sides to focus on x:
\begin{align*}4x + 6 - 3x & = 3x - 3x\x + 6 & = 0.ewline\text{Next, to isolate x, subtract 6 from both sides:}\begin{align*}x + 6 - 6 & = 0 - 6\x & = -6\end{align*}ewline\text{Voila! x is isolated, and we find that x = -6.}
verifying solutions
After finding a solution, it鈥檚 important to verify it to ensure it's correct. To do this, substitute the solution back into the original equation.
In our case, we substitute\( x = -6 \) back into the original equation: \begin{align*} \frac{2(-6) + 3}{-6} & = \frac{3}{2} \end{align*}
The left side simplifies to: \begin{align*} \frac{-12 + 3}{-6} & = \frac{-9}{-6} \end{align*}
Simplifying further, we get: \frac{3}{2}\, which matches the right side of the original equation. Thus, x = -6 is confirmed as the correct solution.
Never skip this step, as it helps confirm that no mistakes were made during the solving process.

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