Question

Solve the proportion. Check for extraneous solutions. $$\frac{8}{x+2}=\frac{3}{x-1}$$

Short answer

The solution to the given proportion is \(x = \frac{14}{5}\), providing it is not an extraneous solution.

Step-by-step solution

Cross Multiply

Multiply \(8\) by \(x - 1\) and \(3\) by \(x + 2\). This yields: \[8(x - 1) = 3(x + 2)\]

Simplify Both Sides of the Equation

Simplify both sides of the equation by distributing. This gives: \[8x - 8 = 3x + 6\].

Adjust the equation to isolate x

Subtract \(3x\)from both sides to obtain: \[5x - 8 = 6\]. Then add \(8\) to both sides of the equation to isolate \(x\). The equation now becomes: \[5x = 14\].

Solve for x

Finally, divide both sides of the equation by \(5\) to solve for \(x\). This gives: \[x = \frac{14}{5}\].

Check for Extraneous Solutions

Substituting \(x = \frac{14}{5}\) back into the original proportion: if \(\frac{8}{\frac{14}{5}+2} = \frac{3}{\frac{14}{5}-1}\), then it is a valid solution, otherwise it is extraneous.