Question

solve.\\(|x + 7| = |x - 2|\select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\a. the solution set is \\(\boxed{}\\). (type an integer or a simplified fraction. use a comma to separate answers as needed.)\\b. the solution is all real numbers.

Explanation:

Step1: Square both sides to eliminate absolute values

To solve the equation \(|x + 7| = |x - 2|\), we can square both sides. Squaring both sides of an equation \(|a| = |b|\) gives \(a^2 = b^2\). So, we have \((x + 7)^2=(x - 2)^2\).

Step2: Expand both sides

Expanding \((x + 7)^2\) using the formula \((a + b)^2=a^2 + 2ab + b^2\), we get \(x^2+14x + 49\). Expanding \((x - 2)^2\) using the formula \((a - b)^2=a^2-2ab + b^2\), we get \(x^2-4x + 4\). So the equation becomes \(x^2+14x + 49=x^2-4x + 4\).

Step3: Simplify the equation

Subtract \(x^2\) from both sides of the equation \(x^2+14x + 49=x^2-4x + 4\). This gives \(14x + 49=-4x + 4\).

Step4: Solve for x

Add \(4x\) to both sides: \(14x+4x + 49=-4x + 4x+ 4\), which simplifies to \(18x + 49 = 4\). Then subtract 49 from both sides: \(18x=4 - 49\), so \(18x=-45\). Divide both sides by 18: \(x=\frac{-45}{18}\), which simplifies to \(x =-\frac{5}{2}\).

Answer:

\(-\frac{5}{2}\)