Solving Absolute-value Equations
Learn how to solve Solving Absolute-value Equations with clear steps and practice problems for Grade 9 algebra. Build confidence solving equations and checking your solutions.
Key Concepts
Property To solve an absolute value equation, begin by isolating the absolute value. Then use the definition of absolute value to write the absolute value equation as two equations. Solve each equation, and write the solution set. Explanation Your first mission is to get the absolute value expression all by itself using inverse operations. Once it's isolated, you crack it open into two separate equations—one for the positive outcome and one for the negative. Solve both of these simpler problems, and you've found all the secret answers! It's like one map leading to two treasures. Examples Solve $4|x+2| 5 = 11$. Isolate: $4|x+2|=16 \rightarrow |x+2|=4$. Split into two equations: $x+2=4$ or $x+2= 4$. The final solutions are $x=2$ or $x= 6$.
Common Questions
How do you solve an absolute-value equation?
Set up two cases: the expression inside equals the positive value, and equals the negative value. Solve both equations and check both answers in the original equation.
Why does an absolute-value equation have two solutions?
Because |x| = c means x could be c or -c — both have the same absolute value. Always consider both cases when solving.
What happens when an absolute-value equation has no solution?
If the equation sets an absolute value equal to a negative number (e.g., |x| = -3), there is no solution since absolute values are always non-negative.