How to Graph Quadratic Inequalities? (+FREE Worksheet!)
To graph a quadratic inequality, graph the corresponding parabola first, using a dashed curve for a strict inequality and a solid one otherwise, then shade the region that satisfies it. Testing a single point off the curve tells you which side to shade.
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Exercises for Graphing Quadratic inequalities
Sketch the graph of each function.
- (color{blue}{y<-2x^2})
- (color{blue}{y≥4x^2})
- (color{blue}{y<-2x^2})
- (color{blue}{y≥4x^2})
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Tutor-style math help
Graph Quadratic Inequalities: what to notice and how to work it
Inequalities skill
Inequalities describe a set of possible values. Solve the boundary like an equation, then decide which side of the boundary makes the statement true.
What to notice first
Watch the comparison sign from the first line to the last. Multiplying or dividing by a negative reverses the direction.
Common student mistake
Do not forget open and closed endpoints. Strict signs use open circles; signs with equals use closed circles.
Key formulas and cues
(a<b)
(ale b)
(text{multiply/divide by a negative} Rightarrow text{reverse the sign})
(|x-a|<b Rightarrow a-b<x<a+b)
A reliable path
- Solve the boundaryTemporarily treat the inequality like an equation.
- Choose the sideUse the sign or test a number if the direction is not obvious.
- Graph the solutionUse the correct endpoint and shade the values that work.
Worked examples
Flip the sign
Example: (-3x>12)
- Divide both sides by -3.
- Reverse the inequality sign.
- Simplify 12 divided by -3.
Answer: (x<-4)
Keep the sign
Example: (x+5le9)
- Subtract 5 from both sides.
- No negative multiplication or division happened.
- Keep the sign direction.
Answer: (xle4)
Try one before moving on
Try: Solve (-2xle10).
Answer: (xge-5). Divide by -2 and flip the sign.
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Graph Quadratic Inequalities: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. Solve (x-2>5).
2. When must the inequality sign flip?
3. (xle3) uses:
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