Isolate Before You Calculate
Isolate the absolute value expression before splitting into two cases in Grade 9 algebra: use inverse operations to get |expression| alone on one side, then apply the two-path method.
Key Concepts
Property In some equations, it is necessary to first isolate the absolute value term before splitting it into two equations. Explanation Before you can split the equation into its two positive and negative paths, you have to get the absolute value expression completely by itself! Think of it like unwrapping a gift. You need to get rid of any numbers multiplied or added outside the bars first. Only then can you properly see what's inside and solve. Examples For $5|x| = 20$, first divide both sides by 5 to get $|x| = 4$. Then you can solve for $x=4$ or $x= 4$. In $4|x 2| = 80$, divide by 4 to get $|x 2| = 20$. Now you are ready to split and solve the two cases. To solve $|x 6| 2 = 2$, first add 2 to both sides to isolate the absolute value, which gives you $|x 6|=0$.
Common Questions
Why must you isolate the absolute value before solving?
The two-path method (setting the inside equal to ±b) only works when the absolute value is alone on one side. If you split too early with other operations mixed in, you will generate incorrect equations and wrong solutions.
How do you solve 3|x - 2| - 5 = 10 using isolate-first strategy?
Isolate the absolute value: add 5 (3|x-2| = 15), then divide by 3 (|x-2| = 5). Now apply two paths: x - 2 = 5 → x = 7, or x - 2 = -5 → x = -3. Solutions are x = 7 and x = -3.
What happens if you try to apply two paths before isolating?
You would incorrectly set 3|x - 2| - 5 = 10 directly as two equations with all the extra operations still present, leading to arithmetic errors. Always simplify to |expression| = positive number first.