The Two-Path Solution
Solve absolute-value equations using the two-path method in Grade 9 algebra: write two separate equations (positive and negative cases) from |A|=b and solve each to find both solutions.
Key Concepts
Property To solve an absolute value equation like $|A| = b$ where $b 0$, you must create and solve two separate linear equations: $A = b$ or $A = b$. Explanation Solving an absolute value equation is like exploring a forked road. Since the expression inside could be positive or negative to start with, you must follow both paths to find all possible answers. Just remember to drop the absolute value bars once you split the equation into its two separate, simpler versions to solve. Examples To solve $|x+3| = 16$, create two paths: $x+3 = 16$ (which gives $x=13$) and $x+3 = 16$ (which gives $x= 19$). Solving $|y 5| = 11$ gives you $y 5=11$ (so $y=16$) and $y 5= 11$ (so $y= 6$). The equation $|2k| = 10$ splits into two possibilities: $2k=10$ and $2k= 10$, so the solutions are $k=5$ or $k= 5$.
Common Questions
What is the two-path method for solving absolute value equations?
For |A| = b where b > 0, write two separate equations: A = b (positive path) and A = -b (negative path). Solve both linear equations independently to find both solutions of the original absolute value equation.
How do you apply the two-path method to solve |2x - 3| = 7?
Path 1: 2x - 3 = 7 → 2x = 10 → x = 5. Path 2: 2x - 3 = -7 → 2x = -4 → x = -2. The two solutions are x = 5 and x = -2. Both should be verified in the original equation.
What happens with the two-path method when b = 0 or b < 0?
If b = 0, both paths give the same equation A = 0, producing one solution. If b < 0, the absolute value equation has no solution because absolute value is always non-negative — it can never equal a negative number.