How many solutions does the following equation have?
$|x+9|=-2$
The problem asks to find the number of real solutions to the equation \[|x+9| = -2.\] The absolute value function, denoted by \(|u|\), is defined for any real number \(u\) as the distance of \(u\) from zero on the number line. By this definition, the absolute value of any real number is always non-negative. That is, for any \(u \in \mathbb{R}\), we have the property \(|u| \geq 0.\)
Given our equation \(|x+9| = -2\), we can substitute \(u = x+9\). The equation becomes \(|u| = -2\). Since we know that \(|u| \geq 0\) for all real numbers \(u\), the expression on the left-hand side is always greater than or equal to zero. However, the right-hand side is -2, which is less than zero. Since a non-negative number cannot equal a negative number, there is no real value of \(u\) (and therefore no real value of \(x\)) that satisfies this equation.
Because the equality \(|x+9| = -2\) contradicts the definition that the absolute value must be non-negative, the equation has no solutions in the set of real numbers. Therefore, the number of real solutions is 0.
The absolute value function is often used to calculate distances or deviations from a target. For example, if you are measuring the deviation of a product's weight from its ideal weight, the deviation is given by the absolute difference. Just as it is impossible for a distance to be negative, it is impossible for an absolute value to be negative. Recognizing these inherent properties saves time when solving equations and helps avoid errors in data analysis.