What is the solution to $|2 x+3|=15$ ?
The given equation is \(|2x + 3| = 15\). The absolute value of an expression, \(|u| = c\), where \(c > 0\), implies that the expression \(u\) itself must be equal to \(c\) or \(-c\). This is because the absolute value measures the distance of a number from zero on the number line, and both \(c\) and \(-c\) are at a distance of \(c\) from zero. Therefore, we can break the given absolute value equation into two separate linear equations:
To find the values of \(x\), we solve each linear equation individually.
For the first equation:
\[2x + 3 = 15\]
Subtract 3 from both sides:
\[2x = 15 - 3\]
\[2x = 12\]
Divide by 2:
\[x = \frac{12}{2} = 6\]
For the second equation:
\[2x + 3 = -15\]
Subtract 3 from both sides:
\[2x = -15 - 3\]
\[2x = -18\]
Divide by 2:
\[x = \frac{-18}{2} = -9\]
Thus, the solutions are \(x = 6\) and \(x = -9\).
We have found that the solutions to the equation \(|2x + 3| = 15\) are \(x = 6\) and \(x = -9\). Looking at the options provided, the correct choice is the one stating \(x = 6\) or \(x = -9\). Therefore, the solution is \(\boxed{x = 6 \text{ or } x = -9}\).
Absolute value equations are essential in real-world scenarios where only the magnitude of a difference matters, regardless of direction. For instance, if a manufacturing machine produces metal rods, the target length might be 100 cm with a tolerance of 5 cm. This is represented by \(|x - 100| \leq 5\). By solving such inequalities or related equations, engineers can determine the range of acceptable lengths for the final product, ensuring quality control in industrial processes.