Solving Absolute Value Equations: Two-Case Method
Solving absolute value equations using the two-case method is a Grade 7 math skill from Yoshiwara Intermediate Algebra. Since |A| = d means A = d or A = -d, students split the equation into two cases and solve each independently, then verify both solutions.
Key Concepts
Property The equation $|ax + b| = c$ (where $c 0$) is equivalent to: $$ax + b = c \quad \text{or} \quad ax + b = c$$.
Examples To solve $|x 3| = 8$, we set up two equations: $x 3 = 8$ or $x 3 = 8$. The solutions are $x = 11$ and $x = 5$. To solve $|2y + 5| = 11$, we set up two equations: $2y + 5 = 11$ or $2y + 5 = 11$. The solutions are $y = 3$ and $y = 8$. To solve $|\frac{z}{3} 1| = 4$, we set up two equations: $\frac{z}{3} 1 = 4$ or $\frac{z}{3} 1 = 4$. The solutions are $z = 15$ and $z = 9$.
Explanation To solve an absolute value equation, you split it into two separate linear equations. This is because the expression inside the absolute value bars could be either positive or negative, and both would result in the same positive value.
Common Questions
What is the two-case method for solving absolute value equations?
If |A| = d (where d > 0), set up two equations: A = d and A = -d. Solve each and check both answers.
How do you solve |2x - 3| = 7?
Case 1: 2x - 3 = 7, so x = 5. Case 2: 2x - 3 = -7, so x = -2. Both solutions are x = 5 and x = -2.
What if |A| = 0?
If |A| = 0, there is exactly one solution: A = 0.
What if |A| = negative number?
If |A| = d where d < 0, there is no solution. Absolute value is never negative.