How to solve absolute value equations

Many students encounter absolute value equations while studying algebra, often feeling overwhelmed by their unique properties. Imagine a high school student grappling with a homework problem that involves finding values of \( x \) in the equation \( |2x – 3| = 5 \). After attempting to follow the typical algebraic rules, they may find themselves stuck, leading to frustration and confusion. This common scenario prompts a search for clarity on how to approach these equations effectively and with confidence. Let’s dive into the steps needed to solve absolute value equations.

To solve an absolute value equation, isolate the absolute value expression and set up two separate equations: one for the positive case and one for the negative case. Then, solve each equation for the variable.

To tackle absolute value equations, start by isolating the absolute value expression on one side of the equation. For example, if you have an equation like \( |x – 4| = 3 \), you want to ensure that \( |x – 4| \) is standing alone. This leads to two scenarios, representing the definition of absolute value: \( x – 4 = 3 \) and \( x – 4 = -3 \).

Next, solve each equation separately. For \( x – 4 = 3 \), add 4 to both sides to get \( x = 7 \). For \( x – 4 = -3 \), again add 4 to both sides, which gives you \( x = 1 \). Thus, the solutions to the equation \( |x – 4| = 3 \) are \( x = 7 \) and \( x = 1 \).

If the absolute value expression is part of a more complex equation, like \( |2x + 1| = 4 \), you would follow the same principle: isolate the absolute value, generate the two scenarios, and solve for \( x \). Remember to check each solution in the original equation to ensure validity, especially in cases where the absolute value is part of a larger problem. This systematic approach will make solving absolute value equations easier and reinforces your algebra skills.