Comprehensive Guide to Solving Quadratic Inequalities in One Variable for Grade 10 Mathematics

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Grade 10 | Mathematics 10 | Term 1 Week 5 Opening
1
Quadratic Inequalities in One Variable
Grade 10 Mathematics | Term 1 Week 5 | 4 Sessions
Big Question: How can we find
all values of x that make a
quadratic expression less than,
greater than, at most, or at
least zero?
Visual cue: roots split the graph and
number line
Instructional visual placeholder: sketch/graph where
the expression is above or below the x-axis.
By the end, you will solve quadratic inequalities using factoring or the quadratic
formula, show solutions on a number line, and express answers in correct
notation.

2
Whole-Lesson Table of Contents
Quadratic Inequalities in One Variable
Ses
sio
n
Actual Session Topic Main Discussion Focus Skill / Output Focus
1
From equations to
inequalities
Standard forms, critical points, number-
line intervals
Identify intervals before solving
2 Solving by factoring Six-step process, sign tests, notation Write and justify solution intervals
3 Solving by quadratic formula
Formula for critical points, same interval
logic
Use formula when factoring is difficult
4
Mixed solving and problem
solving
Choose a method, interpret answers,
reflect
Solve, explain, and apply

3
Big Goals for This Lesson
Understand
•Quadratic inequalities use <, >, ,
≤
or instead of =.
≥
•Critical points divide the number
line into intervals.
•A sign test tells which intervals
make the inequality true.
Do and Show
•Solve by factoring when the
expression is factorable.
•Use the quadratic formula when
factoring is difficult.
•Write answers in inequality,
interval, or set notation and explain
why they work.

4
Essential Question
How do critical points, interval testing, and solution
notation help us describe all possible values that
satisfy a quadratic inequality?
Solving an inequality is not just finding two numbers. It is
finding the range or ranges where the statement is true.

Grade 10 | Mathematics 10 | Term 1 Week 5 Session 1
5
Session 1: From Quadratic Equations to Quadratic
Inequalities
How do roots and signs become intervals?
From Quadratic
Equations to Quadratic
Inequalities
Number line + parabola visual cue
Instructional visual placeholder: sketch/graph
where the expression is above or below the x-axis.
We study visible examples first, discuss the reasoning, analyze three reinforced
examples, then check understanding.

6
Session Roadmap: From Quadratic Equations to
Quadratic Inequalities
• First, we review the important idea for this session.
•Next, we study a visible worked example before analysis.
•Then, we identify the clue: critical points, intervals, signs, or
notation.
•After that, we solve three reinforced examples with different
patterns.
•Finally, we check understanding and prepare the after-class task.

7
Motivation: Inequality Is About Ranges
Real-life idea
•An equation answers: exactly when
are two quantities equal?
•An inequality answers: when is a
quantity below, above, at most, or at
least a limit?
•In real contexts, limits may describe
height, time, distance, budget, or
safe range.
Math connection
•Equation: x² + 4x 5 = 0 gives
−
boundary values.
•Inequality: x² + 4x 5 < 0 asks
−
where the expression is negative.
•The answer may be an interval, not
just one value.

8
Learning Goals: What We Will Learn and Do
• I can solve selected quadratic equations by factoring.
•I can recall the Zero-Factor Property.
•I can define a quadratic inequality in one variable.
•I can identify standard forms using <, >, , and .
≤ ≥
•I can mark critical points and intervals on a number line.

9
Key Words for Quadratic Inequalities
Vocabulary
•Quadratic inequality: an inequality
involving ax² + bx + c and <, >, , or
≤
.
≥
•Critical points: values of x where the
expression equals zero.
•Interval: a continuous part of the
number line.
•Sign test: checking whether the
expression is positive or negative on
an interval.
Remember
•Positive means greater than zero.
•Negative means less than zero.
•≤ and include endpoints when
≥
the expression equals zero.
•< and > exclude endpoints.

10
Visible Review: Solving Quadratic Equations
Solve by factoring:
• x² 2x = 0 x(x 2)=0 x=0 or 2
− → − →
•x² x 6 = 0 (x 3)(x + 2)=0 x=3 or 2
− − → − → −
•2x² + 5x 3 = 0 (2x 1)(x + 3)=0 x=1/2 or 3
− → − → −
Clue: after factoring, each factor can be set to zero because of the
Zero-Factor Property.

11
Concept Explanation: Equation vs. Inequality
Quadratic equation
•Uses an equal sign: ax² + bx + c = 0.
•Usually gives boundary values or
roots.
•Example: x² + 4x 5 = 0 has roots
−
5 and 1.
−
Quadratic inequality
•Uses <, >, , or .
≤ ≥
•Usually gives intervals where the
expression is positive or negative.
•Example: x² + 4x 5 < 0 is true
−
between 5 and 1.
−

12
The Standard Forms We Will Use
• ax² + bx + c < 0 means the expression is negative.
•ax² + bx + c > 0 means the expression is positive.
•ax² + bx + c 0 means the expression is negative or zero.
≤
•ax² + bx + c 0 means the expression is positive or zero.
≥
•In all cases, a, b, and c are real numbers and a ≠ 0.

13
First Visible Example: Critical Points Split the
Number Line
Example: x² + 4x 5 = (x + 5)(x 1)
− −
← →
−5 1
The critical points 5 and 1 divide the number line into three
−
intervals: x < 5, 5 < x < 1, and x > 1. We test one value from
− −
each interval.

14
How the Sign Test Works
• Choose one test value inside
each interval.
•Substitute the test value into
each factor.
•A product with same signs is
positive; different signs is
negative.
•Select the interval that
matches the inequality
symbol.
Interval Test x+5 x 1
− Product
x < 5
− −6 − − +
−5 < x < 1 0 + − −
x > 1 3 + + +
For “< 0,” choose where the product is
negative.

15
Reinforced Example 1: Is It a Quadratic Inequality?
Visible examples
•A. x² 5x + 6 = 0
−
•B. x² 5x + 6 0
− ≥
•C. 3x + 4 < 10
•D. 2x² 5x 3 < 0
− −
Clue and explanation
•B and D are quadratic inequalities
because they have x² and an
inequality symbol.
•A is a quadratic equation because it
uses =.
•C is an inequality, but it is linear,
not quadratic.
•Transfer: check both the highest
power and the symbol.
Quick task: Point to the two quadratic inequalities and explain your clue.

16
Reinforced Example 2: Find Critical Points
Visible example: x² x 6 = (x 3)(x + 2)
− − −
← →
−2 3
• Set each factor equal to zero: x 3 = 0 and x + 2 = 0.
−
•Critical points are x = 3 and x = 2.
−
•Transfer: every factor creates a boundary value for the number line.

17
Reinforced Example 3: Decide Endpoint Inclusion
Visible examples
•x² 5x + 6 < 0
−
•x² 5x + 6 0
− ≤
•x² 5x + 6 > 0
−
•x² 5x + 6 0
− ≥
What to notice
•< and > mean the expression is not
equal to zero, so endpoints are
open.
•≤ and mean zero is allowed, so
≥
endpoints are closed.
•Transfer: circle the inequality
symbol before drawing endpoint
circles.
•Quick task: Say “open” or “closed”
for each example.

18
Watch Out: Do Not Stop at the Roots
Near-miss answer: “x = 5 and x = 1” for x² + 4x 5 < 0
− −
• Why it does not fit: the inequality asks for all x-values that
make the expression negative, not only where it equals zero.
•Correct thinking: roots are boundary values. Use them to
create intervals, then test intervals.
•Reminder: equation roots; inequality intervals or ranges.
→ →

19
Check Your Understanding: Critical Points and
Intervals
•Model item: x² + 2x 8 = (x + 4)(x 2). Critical points are 4 and 2; intervals are
− − −
x < 4, 4 < x < 2, and x > 2.
− −
•Practice 1: Identify critical points for x² + 5x + 6.
•Practice 2: Identify intervals for critical points 3 and 4.
−
•Practice 3: For “ ,” should endpoints be open or closed?
≤
•Practice 4: Explain why an inequality usually gives a range of answers.

20
Remember These Ideas
•A quadratic inequality compares a quadratic expression to
zero using <, >, , or .
≤ ≥
•Critical points come from solving the related equation ax²
+ bx + c = 0.
•Critical points divide the number line into intervals.
•The sign test tells which intervals make the inequality
true.

21
After-Class Challenge: My Number-Line Guide
• Task: Create a one-page mini-guide titled “Roots Become
Intervals.”
•Include one factorable quadratic expression, its critical points,
a number line, and the intervals formed.
•Output: notebook guide or worksheet answer.
•Use arrows, open/closed dots, and a short explanation in your
own words.

22
Session 2: Solving Quadratic Inequalities by
Factoring
How does the sign test select the correct intervals?
Solving Quadratic
Inequalities by Factoring

23
Session Roadmap: Solving Quadratic Inequalities by
Factoring
notation.
patterns.

24
Motivation: From Boundary Values to True Intervals
A factorable inequality lets us use roots as boundaries and
signs as evidence.
• The roots tell where the expression is zero.
•The intervals tell where the expression is positive or
negative.
•The inequality symbol tells which intervals to keep.
•The final answer should be written clearly using notation.

25
Learning Goals: Solving by Factoring
• I can set one side of a quadratic inequality to zero.
•I can factor the quadratic expression and find critical points.
•I can create intervals and use test values.
•I can include or exclude endpoints correctly.
•I can write solutions using inequality notation, interval notation,
or set notation.

26
Six Steps for Factoring Method
• 1. Set one side of the inequality to zero.
•2. Factor the quadratic expression.
•3. Set each factor equal to zero to find critical points.
•4. Create intervals from the critical points.
•5. Use a sign test with one test point per interval.
•6. Write the solution in correct notation.

27
Worked Example 1: Solve x² + 4x 5 < 0
−
Visible work
•x² + 4x 5 < 0
−
•(x + 5)(x 1) < 0
−
•Critical points: 5 and 1
−
•Intervals: x < 5, 5 < x < 1, x > 1
− −
Sign test result
•x = 6: product is positive false
− →
for < 0
•x = 0: product is negative true for
→
< 0
•x = 3: product is positive false for
→
< 0
•Solution: 5 < x < 1 or ( 5, 1)
− −

28
Evidence from the Sign Table
Interval Test x + 5 x 1
− Product For < 0
( , 5)
−∞ − −6 − − + False
( 5, 1)
− 0 + − − True
(1, )
∞ 3 + + + False
← →
−5 1

29
Graph Connection: Why the Middle Interval Works
Graph cue: y = x² + 4x 5
−
• The roots 5 and 1 are where
−
the graph crosses the x-axis.
•For 5 < x < 1, the graph is below
−
the x-axis.
•Below the x-axis means y < 0, so
the expression is negative.
•This matches the sign-test
solution ( 5, 1).
−

30
Reinforced Example 1: Positive Outside the Roots
Solve: x² + 6x > 8
−
Steps and clue
•Standard form: x² + 6x + 8 > 0
•Factor: (x + 4)(x + 2) > 0
− −
•For > 0, keep intervals where the
product is positive.
Result and transfer
•Test values: 5, 3, 3
− −
•Positive on the outside intervals.
•Solution: x < 4 or x > 2
− −
•Interval notation: ( , 4) ( 2,
−∞ − ∪ −
)
∞
•Transfer: for an upward parabola,
positive often appears outside
roots.

31
Reinforced Example 2: Include Endpoints for ≥
Solve: 2x² 5x 3
≥ −
Steps and clue
•Standard form: 2x² 5x + 3 0
− ≥
•Factor: (2x 3)(x 1) 0
− − ≥
•Critical points: x = 1 and x = 3/2
•Use closed endpoints because ≥
allows equality.
Result and transfer
•Sign test shows outside intervals
are true.
•Solution: x 1 or x 3/2
≤ ≥
•Interval notation: ( , 1] [3/2, )
−∞ ∪ ∞
•Transfer: always check whether the
symbol includes equality.

32
Reinforced Example 3: A Middle Interval for <
Solve: x² x < 6
−
Steps and clue
•Standard form: x² x 6 < 0
− −
•Factor: (x 3)(x + 2) < 0
−
−
•Test a value in each interval.
Result and transfer
•Product is negative between the
roots.
•Solution: 2 < x < 3
−
•Interval notation: ( 2, 3)
−
•Quick task: Test x = 0 and explain
why it works.

33
Watch Out: Do Not Forget Standard Form
Near-miss
•Item: x² + 6x > 8
−
•Error: factor x² + 6x and ignore 8.
−
•Problem: the inequality is not set to
zero.
•This gives wrong critical points.
Correct thinking
•Move all terms to one side first.
•x² + 6x + 8 > 0
•Then factor: (x + 4)(x + 2) > 0.
•Reminder: standard form first, then
factor.

34
Check Your Understanding: Factoring Method
• Model item: x² + 5x + 6 > 0 (x + 3)(x + 2) > 0 x < 3 or x > 2.
→ → − −
•Practice 1: Solve x² x < 6.
−
•Practice 2: Solve 3x² > 2x + 5.
•Practice 3: Solve 2x² + 5x 3.
≤
•Practice 4: Explain why endpoints are included in Practice 3.

35
• Set one side of the inequality to zero before factoring.
•Critical points come from the related equation.
•Use one test value for each interval.
•Write the intervals that make the original inequality true.
•Use brackets for included endpoints and parentheses for
excluded endpoints.

36
After-Class Challenge: Six-Step Solution Poster
• Task: Solve one quadratic inequality by factoring using all six
steps.
•Choose from: x² + 5x + 6 > 0, 3x² > 2x + 5, or 5x² > 8x 3.
−
•Show: standard form, factored form, critical points, number line,
sign test, and final notation.
•Output: notebook solution or one-page poster.

37
Session 3: Solving Quadratic Inequalities by the
Quadratic Formula
How do we solve when factoring is not convenient?
Session focus
Solving Quadratic
Inequalities by the
Quadratic Formula

38
Session Roadmap: Solving Quadratic Inequalities by
the Quadratic Formula
notation.
patterns.

39
Motivation: What If Factoring Is Difficult?
Challenge
•Some quadratic expressions are not
easy to factor.
•Guessing roots can waste time and
cause errors.
•We still need critical points before
using interval tests.
Solution
•Use the quadratic formula to find
the critical points.
•Then continue with the same
number-line and sign-test process.
•The method changes only how
critical points are found.

40
Learning Goals: Formula Method
• I can identify a, b, and c in ax² + bx + c = 0.
•I can use x = ( b ± (b² 4ac)) / 2a to find critical points.
− √ −
•I can build intervals from formula roots.
•I can use sign tests to choose correct intervals.
•I can compare formula results with graph or number-line
reasoning.

41
Key Formula and Meaning of a, b, c
For ax² + bx + c = 0, use x = ( b ± (b² 4ac)) / 2a
− √ −
Identify the coefficients
•a is the coefficient of x².
•b is the coefficient of x.
•c is the constant term.
•Always put the expression in
standard form first.
Why it matters
•The formula gives the critical
points.
•These critical points still divide the
number line.
•We still test intervals because the
inequality asks for ranges.

42
Worked Example: Formula Gives the Same Critical
Points
Solve x² + 4x 5 < 0 using the quadratic formula.
−
• a = 1, b = 4, c = 5
−
•x = ( 4 ± (4² 4(1)( 5))) / 2(1)
− √ − −
•x = ( 4 ± 36) / 2 = ( 4 ± 6) / 2
− √ −
•Critical points: x = 5 and x = 1
−
•Use interval testing: solution is 5 < x < 1 or ( 5, 1).
− −

43
Formula Method Still Needs a Sign Test
← →
−5 1
• Critical points only mark where the expression equals zero.
•The sign test tells where the expression is less than zero, greater than
zero, at most zero, or at least zero.
•Transfer: formula roots are not automatically the answer; they are
boundaries.

44
Reinforced Example 1: Solve x² 6x > 5
− −
Visible work
•Standard form: x² 6x 5 > 0
− − −
•Equivalent for easier roots: x² + 6x +
5 < 0
•Factor or formula roots: x = 5 and
−
x = 1
−
•Test intervals around 5 and 1.
− −
Result and explanation
•The expression is true between the
roots.
•Solution: 5 < x < 1
− −
•Interval notation: ( 5, 1)
− −
•Transfer: multiplying by 1
−
reverses the inequality symbol.

45
Reinforced Example 2: Solve x² + 10x 16
− ≤
Visible work
•Standard form: x² + 10x 16 0
− − ≤
•Equivalent: x² 10x + 16 0
− ≥
•Roots: x = 2 and x = 8
•Use sign test and include endpoints
for .
≥
Result and explanation
•The true intervals are outside the
roots.
•Solution: x 2 or x 8
≤ ≥
•Interval notation: ( , 2] [8, )
−∞ ∪ ∞
•Transfer: after changing signs,
keep the reversed inequality in
mind.

46
Reinforced Example 3: Radical Critical Points
Solve: x² 2x + 1
≤
Formula roots
•Standard form: x² 2x 1 0
− − ≤
•a = 1, b = 2, c = 1
− −
•x = (2 ± (4 + 4)) / 2
√
•x = (2 ± 8) / 2 = 1 ± 2
√ √
Solution and transfer
•Use intervals around 1 2 and 1
− √
+ 2.
√
•Since the parabola opens upward
and asks 0, keep the middle
≤
interval.
•Solution: 1 2 x 1 + 2
− √ ≤ ≤ √
•Interval: [1 2, 1 + 2]
− √ √

47
Watch Out: Sign Errors in b² 4ac
−
Common error
•For x² 2x 1 = 0, some learners
− −
write b² 4ac = ( 2)² 4(1)(1).
− − −
•This changes c from 1 to +1.
−
•The resulting roots are wrong.
Correct thinking
•Read c with its sign: c = 1.
−
•b² 4ac = ( 2)² 4(1)( 1) = 8.
− − − −
•Reminder: box a, b, and c before
substitution.

48
Check Your Understanding: Formula Method
• Model item: x² 2x + 1 x² 2x 1 0 roots 1 ± 2 [1 2, 1 + 2].
≤ → − − ≤ → √ → − √ √
•Practice 1: Identify a, b, c in x² 5x 6 < 0.
− −
•Practice 2: Find the critical points of x² 5x 3.
≤ −
•Practice 3: Should endpoints be included for ?
≤
•Practice 4: Explain why the quadratic formula does not replace the sign test.

49
• Use the quadratic formula when factoring is difficult or not
practical.
•Put the inequality in standard form before identifying a, b,
and c.
•Formula roots are critical points, not automatically the final
solution.
•After finding critical points, still create intervals and test
values.

50
After-Class Challenge: Formula Card
• Task: Make a formula reminder card for solving quadratic
inequalities.
•Include: the formula, how to identify a, b, c, and one sample
item.
•Show the critical points and the final interval solution.
•Add one personal warning about a mistake to avoid.

51
Session 4: Mixed Solving, Notation, and Problem
Solving
How do we choose a method and interpret the solution?
Session focus
Mixed Solving, Notation,
and Problem Solving

52
Session Roadmap: Mixed Solving, Notation, and
Problem Solving
notation.
patterns.

53
Motivation: Choose the Best Tool
When factoring is efficient
•The expression factors cleanly.
•The roots are easy to see.
•You can complete the sign test
quickly.
When formula is helpful
•Factoring is difficult or not obvious.
•Roots may involve radicals.
•You still need intervals, signs, and
notation after finding roots.

54
Learning Goals: Mixed Method and Interpretation
• I can decide whether factoring or the quadratic formula is
efficient.
•I can solve mixed quadratic inequalities accurately.
•I can express answers in correct notation.
•I can interpret intervals in simple problem contexts.
•I can justify why a selected interval is the solution.

55
Notation Review: Three Ways to Show the Same
Answer
Example: outside intervals
•Inequality notation: x < 4 or x >
−
2
−
•Interval notation: ( , 4) ( 2,
−∞ − ∪ −
)
∞
•Set notation: {x | x < 4 or x > 2}
− −
What the symbols mean
•Parentheses mean endpoint not
included.
•Brackets mean endpoint included.
•∪ means union: combine two
solution intervals.
•∞ is never written with a bracket.

56
Reinforced Example 1: Solve x² 5x > 0
−
Visible work
•Factor: x(x 5) > 0
−
•Test intervals: x < 0, 0 < x < 5, x > 5
•Keep where product is positive.
•Solution: x < 0 or x > 5
•Interval notation: ( , 0) (5, )
−∞ ∪ ∞
•Transfer: outside intervals can
happen when the expression opens
upward and asks for > 0.
•Quick task: test x = 1 to explain why
the middle interval is false.

57
Reinforced Example 2: Solve 4x² 8x 5
− ≤
Visible work
•Standard form: 4x² 8x 5 0
− − ≤
•Critical points: x = 1/2 and x = 5/2
−
•Use , so endpoints are included.
≤
•Test intervals around 1/2 and 5/2.
−
•The true interval is between the
roots.
•Solution: 1/2 x 5/2
− ≤ ≤
•Interval notation: [ 1/2, 5/2]
−
•Transfer: when a parabola opens
upward, usually keeps the middle
≤
region.

58
Reinforced Example 3: Product Form Already Given
Visible work
•Solve: 9x(x + 1) 10
≥
•Standard form: 9x² + 9x 10 0
− ≥
•Critical points: x = 5/3 and x = 2/3
−
•Use , so include endpoints.
≥
•True intervals are outside the
roots.
•Solution: x 5/3 or x 2/3
≤ − ≥
•Interval notation: ( , 5/3]
−∞ − ∪
[2/3, )
∞
•Quick task: Why does x = 0 not
work?

59
Problem Solving Model: Interpreting a Range
A ball’s height is modeled by h(t) = t² + 6t + 7. For what times is the height
−
at least 12 units?
• Translate: t² + 6t + 7 12
− ≥
•Standard form: t² + 6t 5 0, or t² 6t + 5 0
− − ≥ − ≤
•Factor: (t 1)(t 5) 0
− − ≤
•Solution: 1 t 5
≤ ≤
•Interpretation: the ball is at least 12 units high from t = 1 to t = 5.

60
Watch Out: Final Answer Must Match the Question
Mistake
•Solve correctly but write only roots.
•Write interval notation but forget to
interpret in context.
•Use parentheses when the problem
says “at least” or “at most.”
Correct thinking
•Question asks “when,” “for what
values,” or “range.”
•Use solution intervals and then
explain in words.
•At least / at most usually means
include endpoints if equality is
allowed.

61
Check Your Understanding: Mixed Method Exit
Practice
• Model item: x² 5x < 6 x² 5x 6 < 0 (x 6)(x + 1) < 0 1 < x < 6.
− → − − → − → −
•Practice 1: Solve x² 5x > 0.
−
•Practice 2: Solve x² 5x < 6.
−
•Practice 3: For or , are critical points included?
≥ ≤
•Practice 4: Write one sentence explaining why we test values from each
interval.

62
• Choose factoring when it is efficient; use the quadratic
formula when factoring is not convenient.
•Both methods require critical points, intervals, and sign
tests.
•Write final answers using correct notation.
•In context, explain what the interval means, not only the
algebraic answer.

63
After-Class Challenge: Mixed Practice Reflection
• Task: Solve two mixed quadratic inequalities and label your
method.
•For each item, include critical points, interval test, and final
notation.
•Then answer: “What is one mistake I should avoid when solving
quadratic inequalities?”
•Output: worksheet or notebook entry.

Grade 10 | Mathematics 10 | Term 1 Week 5 Closing
64
Whole-Lesson Review
• Session 1: We connected quadratic equations to quadratic inequalities
and learned that roots become interval boundaries.
•Session 2: We solved quadratic inequalities by factoring and sign testing.
•Session 3: We used the quadratic formula to find critical points when
factoring is difficult.
•Session 4: We solved mixed items, used notation correctly, and
interpreted solution ranges.
•Essential answer: critical points, intervals, and signs work together to
describe all values that make an inequality true.

Grade 10 | Mathematics 10 | Term 1 Week 5 Appendix
65
Answer Key / Teacher Cues: Session 1
• Practice 1: x² + 5x + 6 = (x + 2)(x + 3); critical points 3 and 2.
− −
•Practice 2: intervals are x < 3, 3 < x < 4, and x > 4.
− −
•Practice 3: for , endpoints are closed/included.
≤
•Practice 4: an inequality asks for all x-values that make the statement true,
so the answer is usually a range.

66
• x² x < 6 x² x 6 < 0 2 < x < 3.
− → − − → −
•3x² > 2x + 5 3x² 2x 5 > 0 x < 1 or x > 5/3.
→ − − → −
•2x² + 5x 3 2x² + 5x 3 0 3 x 1/2.
≤ → − ≤ → − ≤ ≤
•Endpoints are included in the third item because the symbol is .
≤

67
• a, b, c in x² 5x 6 < 0: a = 1, b = 5, c = 6.
− − − −
•x² 5x 3 x² 5x + 3 0; roots are (5 13)/2 and (5 + 13)/2; solution is
≤ − → − ≤ − √ √
[(5 13)/2, (5 + 13)/2].
− √ √
•For , endpoints are included.
≤
•Formula finds critical points; sign testing still chooses the true intervals.

68
• x² 5x > 0 x(x 5) > 0 x < 0 or x > 5; ( , 0) (5, ).
− → − → −∞ ∪ ∞
•x² 5x < 6 x² 5x 6 < 0 1 < x < 6; ( 1, 6).
− → − − → − −
•For or , critical points are included.
≥ ≤
•We test one value from each interval because the sign of the quadratic
expression is constant within each interval unless it crosses a critical point.

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Additional Mixed Practice Answers
• 5x² > 8x 3 x < 3/5 or x > 1.
− →
•−x² 6x > 5 5 < x < 1.
− → − −
•−x² + 10x 16 x 2 or x 8.
≤ → ≤ ≥
•x² 2x + 1 1 2 x 1 + 2.
≤ → − √ ≤ ≤ √
•9x(x + 1) 10 x 5/3 or x 2/3.
≥ → ≤ − ≥

Comprehensive Guide to Solving Quadratic Inequalities in One Variable for Grade 10 Mathematics

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