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Quadratic Inequalities
Grade 7/8 Quadratic Inequalities Solve quadratic inequalities If you have any questions regarding these resources or come across any errors, please contact
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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
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Key Vocabulary Inequality Solve/solution
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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
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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
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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
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Linear inequalities 3 < x < 8 3 ≤ x < 8 3 < x ≤ 8
1 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
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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
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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
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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
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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
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Examination style question
Linear inequalities Examination style question 2n < 11 n < 5.5 5
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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
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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
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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
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Exam Questions – Specimen Papers
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Exam Questions – Specimen Papers
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Exam Questions – Specimen Papers
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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
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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
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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?
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Exam Questions – Specimen Papers
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