MathematicsChapter 5: Linear Equations and Inequalities
Radical Equations and Absolute Value
A radical equation has the unknown under a root.
Definition
A radical equation has the unknown under a root. Isolate the root, then square both sides, then check in the original because squaring can add extra roots. Absolute value |x| is the distance from 0, so |x| = 3 means x = 3 or x = -3. |x| is never negative.
Formula
x = 3 or x
Relationship
- |x| = distance
Overview
Two extra equation types:
Radicals
sqrt(x) = 4 -> x = 16, and 16 works. sqrt(x) = -1 has no real solution. If sqrt(x-1) = x-3, square to x-1 = (x-3)^2, solve the quadratic, then reject any root that makes the root negative or fails the original.
Examples sqrt(2x+1)=3 -> 2x+1=9 · Reject a root that makes sqrt negative · Do not square only one side
Absolute Value
|ax+b| = c with c>=0 splits into ax+b = c or ax+b = -c. If c<0, no solution. Inequalities: |x| < 2 is -2 < x < 2. |x| > 2 is x<-2 or x>2. Draw a number line.
Examples |x-1|=5 -> x=6 or x=-4 · |x|<3 the open segment · Do not write |x| = -5
Explanation
Linear inequalities were earlier pages. Absolute-value inequalities are the same idea plus a split. Shade and use open/closed ends as the inequality asks.
In simple words
Squaring is not an equivalent step unless both sides are non-negative. That is why the check is not optional.
Key points
- isolate then square
- check extra roots
- |x| = distance
- check in the original
- split into two
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Radical Equations and Absolute Value
Sticky Notes · Quick Revision
Core Idea
- A radical equation has the unknown under a root.
Key Points
- isolate then square
- check extra roots
- |x| = distance
Simple Check
- Explain radical equations and absolute value in your own words and connect it to one chapter example.
How to Answer
- Define the idea, show the main rule or evidence, apply it, and check the final statement.
Common Mistake
- Squaring is not an equivalent step unless both sides are non-negative. That is why the check is not optional.
FBISE Exam Tip
- Use the exact term, show working where needed, and include units or labelled evidence.
60-second revision
- isolate then square
- check extra roots
- |x| = distance
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