1. Quadratic Inequalities
Grade 7/8
Quadratic Inequalities
Solve quadratic inequalities
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2. Lesson Plan Lesson Overview Progression of Learning
Objective(s)
Solve quadratic inequalities
Grade
7/8
Prior Knowledge
Solve linear inequalities; represent the results on a number line (a recap of this is included in slides 4-10).
Duration
65 minutes (20 minutes is recap of linear inequalities and can be omitted if desired)
Resources
Slides 19 onwards are printable versions of some of the earlier slides.
Equipment
Progression of Learning
What are the students learning?
How are the students learning? (Activities & Differentiation)
Linear inequalities; recap about using number lines, finding integer solutions and algebraic solutions.
Initial slides show number lines and “graphical” solutions.
Slide 10 introduces the process of simplifying/solving an inequality. PRINT SLIDE 19
Differentiated practice. PRINT SLIDE 20 20
Quadratic inequalities; integer solutions, solutions shown on a number line, algebraic solutions.
Reasoning
First slide shows how x2 ≥ 9 generates two sets of solutions. THREE BUTTONS TO REVEAL; GREEN BUTTONS NEED TWO PRESSES EACH TO REVEAL BOTH. CLICK ON AN INDIVIDUAL INTEGER TO REVEAL WHETHER IT IS (HIGHLIGHTED YELLOW) OR IS NOT (FADES TO GREY) PART OF THE SOLUTION Following slide compares the effect of changing x2 ≥ 9 into x2 > 9, x2 ≤ 9 or x2 < 9. USE THE RED/GREY BUTTONS TO SELECT 15
Differentiated practice. PRINT SLIDE 21
Gold section is a reasoning task, based on “reverse question”.
Solving quadratic inequalities in exam questions (from specimen papers)
PRINT SLIDE 22. This includes 3 exam questions related to objective. Students need to use notes from lesson to answer the questions. Ensure that all steps are shown. Relate to mark scheme to show how the marks are allocated using slides 16 to 18. 10
Next Steps
Assessment
PLC/Reformed Specification/Target 7/Algebra/Solving quadratic inequalities

3. Key Vocabulary
Inequality Solve/solution

4. Linear inequalities x > 1 x ≥ 1 x < 1 x ≤ 1 -1 1 2 3 4 5 6 7 8 9
Choose the inequality that matches the arrow on the number line. -1 1 2 3 4 5 6 7 8 9
x > 1
x ≥ 1
x < 1
x ≤ 1
Go straight to quadratic inequalities
Click on an expression. If it is correct it will turn red; otherwise it will fade to grey

5. Choose the inequality that matches the arrow on the number line.
Linear inequalities
Choose the inequality that matches the arrow on the number line. 3 4 5 6 7 8 9 10 11 12 13
x > 6
x ≥ 6
x < 6
x ≤ 6
Click on an expression. If it is correct it will turn red; otherwise it will fade to grey

6. Choose the inequality that matches the arrow on the number line.
Linear inequalities
Choose the inequality that matches the arrow on the number line. -3 -2 -1 1 2 3 4 5 6 7
-1 < x < 5
-1 ≤ x < 5
-1 < x ≤ 5
-1 ≤ x ≤ 5
Click on an expression. If it is correct it will turn red; otherwise it will fade to grey. Click the slide to highlight the integers

7. Linear inequalities 3 < x < 8 3 ≤ x < 8 3 < x ≤ 81 2 3 4 5 6 7 8 9 10 11
3 < x < 8
3 ≤ x < 8
3 < x ≤ 8
3 ≤ x ≤ 8
Click on an expression. If it is correct it will turn red; otherwise it will fade to grey. Click the slide to highlight the integers

8. Choose the inequality that matches the arrow on the number line.
Linear inequalities
Choose the inequality that matches the arrow on the number line. 1 2 3 4 5 6 7 8 9 10
1 < x < 8
1 ≤ x < 8
1 < x ≤ 8
1 ≤ x ≤ 8
Click on an expression. If it is correct it will turn red; otherwise it will fade to grey. Click the slide to highlight the integers

9. Choose the inequality that matches the arrow on the number line.
Linear inequalities
Choose the inequality that matches the arrow on the number line. -3 -2 -1 1 2 3 4 5 6 7
0 < x < 6
0 ≤ x < 6
0 < x ≤ 6
0 ≤ x ≤ 6
Click on an expression. If it is correct it will turn red; otherwise it will fade to grey. Click the slide to highlight the integers

10. Which of these give the same solution on the number line?
Linear inequalities
Which of these give the same solution on the number line?
x ≥ 3
x + 5 ≥ 10
3 ≤ x
5 ≤ 2x + 1
x + 4 ≥ 8
5 ≤ 2x − 1
2x ≥ 6
2x + 5 ≤ 3x
3x − 6 ≥ x
x + 1 ≥ 4
5x ≥ 20
2x + 1 ≥ 5
x − 1 ≥ 4
3x + 1 ≥ 10 -2 -1 1 2 3 4 5 6 7 8
Click on an expression. If it matches x ≥ 3, it will turn green; otherwise it will turn amber

11. Find the solutions of these linear inequalities
BRONZE
SILVER
GOLD
For the bronze and silver questions, find the inequality in the form
x > 4, 3 < x ≤ 7, etc, and show your solution on a number line
Click “Bronze” “Silver” or “Gold” to see the algebraic solutions
For the silver and gold questions, where it is possible to do so, write down a list of all the integers that satisfy the inequality
2x > 8
10x ≤ 70
2x + 3 > 15
3x − 7 ≥ 5
3x − 6 > 0
4x − 2 ≤ 10
3 < x ≤ 7
3 ≤ x < 7
6 < 3x < 18
−8 < 2x ≤ 6
3 ≤ x + 1 ≤ 8
−11 < x − 4 ≤ −9
3x + 1 < x + 9
2x + 10 ≥ 4x
x > 3x + 4
5x ≤ 3x + 1
9x < 3x − 4
− 2 < 3x ≤ 7
x > 4
x ≤ 7
x + > 6
x ≥ 4
x > 2
x ≤ 3
3 < x ≤ 7
3 ≤ x < 7
2 < x < 6
−4 < x ≤ 3
2 ≤ x ≤ 7
−7 < x ≤ −5
x < 4
x ≤ 5 (or 5 ≥ x)
x < −2 (or −2 > x)
x ≤ ½
x < −⅔
−⅔ < x ≤ 2⅓

12. Examination style question
Linear inequalities
Examination style question
2n < 11
n < 5.5 5

13. Quadratic inequalities
Which integers on this number line give x2 ≥ 9 ?
Solve the inequality x2 ≥ 9 -5 -4 -3 -2 -1 1 2 3 4 5
x ≤ −3
x ≥ 3
Reminder…
Show all integers
Use a “filled in” circle to indicate that the end point can be included (normally for ≤ or ≥)
Show number line
Algebraic solutions
Use an “empty” circle to indicate that the end point can not be included (normally for < or >)
Click on an integer. If it satisfies x2 ≥ 9, it will turn yellow; otherwise it will fade to grey. Press each green button twice for full answer

14. Click an inequality to see solutions
Quadratic inequalities -5 -4 -3 -2 -1 1 2 3 4 5
x < −3
x ≤ −3
−3 ≤ x ≤ −3
−3 < x < −3
x > 3
x ≥ 3
x2 ≤ 9
x2 ≥ 9
x2 < 9
x2 > 9
Click an inequality to see solutions

15. Find the solutions of these quadratic inequalities
BRONZE
SILVER
GOLD
For the bronze and silver questions, solve algebraically and show your solution on a number line. Where it is possible to do so, write down a list of all the integers that satisfy the inequality.
Extension: reasoning question
x2 > 16
x2 ≤ 25
x2 ≥ 4
x2 < 36
x2 ≥ 36
x2 > 1
x2 < 1
x2 > 0
16 > x2
4 ≤ x2
x2 + 5 ≥ 9
x2 − 1 < 8
2x2 ≥ 50
3x2 > 48
4 < x2 < 49
9 ≤ x2 < 16
What are the integers that satisfy the inequalities −4 < x ≤ 3 and x2 < 16?
(You should find that both sets of integers are the same.)
How many different inequalities can you find for which the set of integers is −4, −3, −2, −1, 0, 1, 2, 3, 4?
x < −4 or x > 4
−5 ≤ x ≤ 5
x ≤ −2 or x ≥ 2
−6 < x < 6
x ≤ −6 or x ≥ 6
x < −1 or x > 1
−1 < x < 1
x ≠ 0
−4 < x < 4
x ≤ −2 or x ≥ 2
x < −3 or x > 3
x ≤ −5 or x ≥ 5
x ≤ −4 or x ≥ 4
−7 < x < −2 or 2 < x < 7
−4 < x ≤ −3 or 3 ≤ x < 4
Linear: −5 < x < 5, −4 ≤ x ≤ 4, −5 < x ≤ 4, −4 ≤ x < 5; Quadratic: x2 ≤ 16, x2 < 25;
How about variations on…
−4.5 < x < 4.5, −4.5 ≤ x ≤ 4.5, −9 < 2x < 9;
x2 ≤ 20, x2 < 20, 2x2 < 40 ?
Click “Bronze” “Silver” or “Gold” to see the algebraic solutions

16. Exam Questions – Specimen Papers

17. Exam Questions – Specimen Papers

18. Exam Questions – Specimen Papers

19. Which of these give the same solution on the number line?
Linear inequalities
Which of these give the same solution on the number line?
x ≥ 3
x + 5 ≥ 10
3 ≤ x
5 ≤ 2x + 1
x + 4 ≥ 8
5 ≤ 2x − 1
2x ≥ 6
2x + 5 ≤ 3x
3x − 6 ≥ x
x + 1 ≥ 4
5x ≥ 20
2x + 1 ≥ 5
x − 1 ≥ 4
3x + 1 ≥ 10 -2 -1 1 2 3 4 5 6 7 8

20. Find the solutions of these linear inequalities
BRONZE
SILVER
GOLD
For the bronze and silver questions, find the inequality in the form
x > 4, 3 < x ≤ 7, etc, and show your solution on a number line
For the silver and gold questions, where it is possible to do so, write down a list of all the integers that satisfy the inequality
2x > 8
10x ≤ 70
2x + 3 > 15
3x − 7 ≥ 5
3x − 6 > 0
4x − 2 ≤ 10
3 < x ≤ 7
3 ≤ x < 7
6 < 3x < 18
−8 < 2x ≤ 6
3 ≤ x + 1 ≤ 8
−11 < x − 4 ≤ −9
3x + 1 < x + 9
2x + 10 ≥ 4x
x > 3x + 4
5x ≤ 3x + 1
9x < 3x − 4
− 2 < 3x ≤ 7

21. Find the solutions of these quadratic inequalities
BRONZE
SILVER
GOLD
For the bronze and silver questions, solve algebraically and show your solution on a number line. Where it is possible to do so, write down a list of all the integers that satisfy the inequality.
Extension: reasoning question
x2 > 16
x2 ≤ 25
x2 ≥ 4
x2 < 36
x2 ≥ 36
x2 > 1
x2 < 1
x2 > 0
16 > x2
4 ≤ x2
x2 + 5 ≥ 9
x2 − 1 < 8
2x2 ≥ 50
3x2 > 48
4 < x2 < 49
9 ≤ x2 < 16
What are the integers that satisfy the inequalities −4 < x ≤ 3 and x2 < 16?
(You should find that both sets of integers are the same.)
How many different inequalities can you find for which the set of integers is −4, −3, −2, −1, 0, 1, 2, 3, 4?

22. Exam Questions – Specimen Papers