Question

1. solve: \\(\frac{x + 2}{x - 3} = 4\\).

Explanation:

Step1: Eliminate the denominator

Multiply both sides of the equation \(\frac{x + 2}{x - 3}=4\) by \(x - 3\) (assuming \(x
eq3\) to avoid division by zero) to get \(x + 2 = 4(x - 3)\).

Step2: Expand the right - hand side

Expand \(4(x - 3)\) using the distributive property \(a(b - c)=ab - ac\), where \(a = 4\), \(b=x\) and \(c = 3\). So we have \(x+2=4x-12\).

Step3: Move the terms with \(x\) to one side

Subtract \(x\) from both sides: \(x + 2-x=4x-12 - x\), which simplifies to \(2 = 3x-12\).

Step4: Solve for \(x\)

Add 12 to both sides: \(2 + 12=3x-12 + 12\), so \(14 = 3x\). Then divide both sides by 3: \(x=\frac{14}{3}\).

Step5: Check the solution

Substitute \(x = \frac{14}{3}\) into the original denominator \(x-3=\frac{14}{3}-3=\frac{14 - 9}{3}=\frac{5}{3}
eq0\). And the left - hand side \(\frac{\frac{14}{3}+2}{\frac{14}{3}-3}=\frac{\frac{14 + 6}{3}}{\frac{14 - 9}{3}}=\frac{\frac{20}{3}}{\frac{5}{3}} = 4\), which is equal to the right - hand side.

Answer:

\(x=\frac{14}{3}\)