Solving Absolute Value Equations
Solve absolute value equations in Grade 10 algebra by splitting into two cases, then checking solutions against the original equation to eliminate extraneous answers.
Key Concepts
To solve an absolute value equation like $|x a| = k$, you must create and solve two separate cases derived from the original equation: $x a = k$ or $x a = k$.
To solve $|x 3| = 9$, you must solve both $x 3 = 9$ (which gives $x=12$) and $x 3 = 9$ (which gives $x= 6$). To solve $|2y + 5| = 11$, you solve $2y + 5 = 11$ (getting $y=3$) and $2y + 5 = 11$ (getting $y= 8$).
Since absolute value makes everything positive, the expression inside could have started as either positive or negative. Imagine a mystery where the clue is 'the absolute value is 5.' The original number could have been $5$ or $ 5$. We have to solve for both possibilities to crack the case and find all the hidden solutions!
Common Questions
How do you solve |2x - 3| = 7?
Split into two cases: 2x-3=7 giving x=5, and 2x-3=-7 giving x=-2. Check both in the original equation: |2(5)-3|=7 ✓ and |2(-2)-3|=7 ✓.
Why must you check solutions to absolute value equations?
Extraneous solutions can appear when setting up the two cases. Substituting back into the original equation confirms which solutions are valid.
What does |x - a| = b mean geometrically?
It means x is exactly b units away from a on the number line. This gives two solutions: x = a + b and x = a - b.