Problem 94 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$2 x+5=5 x+2$$

Step-by-Step Solution

Answer

The original statement is false. The true statement should specify the equation is only true for \(x = -1\).

Step 1: Simplify the equation

First, we can simplify by grouping the like terms together to get \(3 x = -3\)

Step 2: Solve for \(x\)

Divide both sides of the equation by 3 to solve for \(x\): \(x = -1.\)

Step 3: Verify the solution

Substitute the value of \(x\) back into the original equation to verify the result. Substituting \(x = -1\) into the original equation \(2x+5 = 5x + 2\) gives \(2*-1+5\) on the left side and \(5*-1+2\) on the right side. Both simplify to 3, so the equation holds true with \(x = -1\). However, initially we assumed that \(2x+5\) is equal to \(5x+2\) for all \(x\), which is not the case. Therefore, the original statement is false, and the true statement is \(2x + 5 = 5x + 2\) only when \(x = -1\).