Solve absolute value equations

Learning Objectives Define absolute value as a measure of distance from zero on a number line. Solve linear absolute value equations of the form |ax + b| = c. Solve absolute value equations with variables on both sides of the equation, such as |ax + b| = cx + d. Solve equations involving two absolute value expressions, such as |ax + b| = |cx + d|. Identify and discard extraneous solutions by checking all potential solutions in the original equation. Model real-world scenarios involving tolerance or margin of error using absolute value equations. How can a GPS tell you you're 50 feet from your destination, without caring if you're 50 feet north, south, east, or west? 🤔 That's the power of absolute value! This tutorial will guide you through the process of sol...

TermDefinitionExample Absolute ValueThe distance of a number from zero on the number line. Since distance cannot be negative, the absolute value of a number is always non-negative.|-7| = 7 and |7| = 7. Both -7 and 7 are 7 units away from 0. Absolute Value EquationAn equation that includes an absolute value expression involving a variable.|2x - 3| = 11 Isolating the Absolute ValueThe essential first step in solving an absolute value equation, where you use inverse operations to get the absolute value expression by itself on one side of the equation.To solve 3|x + 2| - 1 = 8, you first add 1 to both sides, then divide by 3 to get |x + 2| = 3. Case AnalysisThe process of splitting an absolute value equation into two separate equations (one positive case, one negative case) to find all possib...

The Fundamental Rule If \|X\| = c and c \geq 0, then X = c or X = -c. This is the primary rule for solving most absolute value equations. Once the absolute value is isolated, you create two separate equations to solve. The No Solution Rule If \|X\| = c and c < 0, there is no solution. Use this as a quick check. Since absolute value represents distance, it can never be negative. If an absolute value expression is equal to a negative number, the equation has no solution. The Two Absolute Values Rule If \|X\| = \|Y\|, then X = Y or X = -Y. When an equation has a single absolute value expression on each side, you can set up two cases: the expressions are equal, or the expressions are opposites.