Absolute-value equation
Solve absolute value equations by splitting into two cases: positive and negative. Use distance interpretation to master Grade 9 absolute value problems.
Key Concepts
Property An equation that has one or more absolute value expressions is called an absolute value equation. The absolute value of a number $n$ is the distance from $n$ to 0 on a number line. Explanation Think of absolute value as a 'distance detector'—it only measures how far a number is from zero, ignoring whether it's positive or negative. So, an equation like $|x| = 5$ simply asks, 'Which numbers are exactly 5 steps away from zero?' This always leads to two possibilities, one on each side of zero. Examples If $|x| = 7$, it means $x$ is 7 units from 0, so $x = 7$ or $x = 7$. The expression $|y+2| = 6$ means the entire quantity $(y+2)$ must be 6 units away from 0. The solution set for $|z| = 13$ is $\{13, 13\}$ because both numbers are 13 units from 0.
Common Questions
How do you solve an absolute value equation like |2x-3| = 7?
Split into two: 2x-3 = 7 and 2x-3 = -7. Solve each: x=5 and x=-2. Verify both solutions in the original equation.
Why does an absolute value equation produce two solutions?
Absolute value measures distance from zero, so both a positive and negative value can have the same absolute value. |5| = 5 and |-5| = 5.
When does an absolute value equation have no solution?
When the absolute value equals a negative number, like |x+1| = -3. Since absolute value is always at least 0, no real x can satisfy this.