Question

Solve the equation. \(|2 x-5|+7=16\)

Short answer

The solutions for the equation \(|2 x-5|+7=16\) are \(x=7\) and \(x=-2\).

Step-by-step solution

Isolate the absolute value term

To begin with, the target is to isolate the absolute value term on one side of the equation. This means getting rid of '+7' on the left side of the equation. This can be achieved by subtracting '7' from both sides. This operation then gives the equation \(|2 x-5|=16-7\), which simplifies to \(|2x-5|=9\).

Split the equation

Since absolute value measures the 'distance' of a number from zero on the number line, it could either be positive or negative. So an equation that contains an absolute value can be split into two separate equations. We then get \(2x-5=9\) and \(2x-5=-9\).

Solve for X in each equation

Now comes the part where the value of 'x' is determined from each equation separately. To solve for 'x' in the first equation, add '5' to both sides of the equation \(2x=9+5\) which simplifies to \(2x=14\), then divide both sides of the equation by '2' to find the value of 'x' which is \(x=\frac{14}{2}\) or \(x=7\). In the second equation, add '5' to both sides of the equation \(2x=-9+5\) which simplifies to \(2x=-4\), then divide both sides of the equation by '2' to solve for 'x'. This results to \(x=\frac{-4}{2}\) or \(x=-2\).