Question

$\frac{1}{x + 2} = 5$

Explanation:

Step1: Cross - multiply the equation

Given the equation \(\frac{1}{x + 2}=5\), we can cross - multiply. Cross - multiplying means multiplying both sides of the equation by \(x + 2\) and also by \(1\) (implicitly) to get rid of the fraction. So we have \(1=5\times(x + 2)\).

Step2: Expand the right - hand side

Expand \(5\times(x + 2)\) using the distributive property \(a(b + c)=ab+ac\), where \(a = 5\), \(b=x\) and \(c = 2\). So \(5\times(x + 2)=5x+10\). Now our equation is \(1 = 5x+10\).

Step3: Solve for \(x\)

Subtract \(10\) from both sides of the equation: \(1-10=5x+10 - 10\). This simplifies to \(-9 = 5x\). Then divide both sides by \(5\): \(x=\frac{-9}{5}=-1.8\) (or we can keep it as a fraction \(x =-\frac{9}{5}\)). We should also check if this solution makes the original denominator non - zero. When \(x=-\frac{9}{5}\), \(x + 2=-\frac{9}{5}+2=-\frac{9}{5}+\frac{10}{5}=\frac{1}{5}
eq0\), so the solution is valid.

Answer:

\(x =-\frac{9}{5}\) (or \(x=-1.8\))