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Uploaded byWaleedAbuelela
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The document provides an interview summary of a lesson on solving various types of inequalities: 1) The lesson covers linear, quadratic, and rational inequalities, explaining the steps to solve each type. 2) Examples are worked through demonstrating how to identify intervals on the number line and use test values to determine the solution set of inequalities. 3) Applications involving break-even points and projectile motion are presented to show real-world examples.

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1.7 - 1Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 Equations and Inequalities Interview with .... Solving Inequalities By Waleed Abuelela March 12, 2019
By the end of this lesson you should know how to solve • Linear Inequalities • Quadratic Inequalities • Rational Inequalities
What do you know about inequalities? What are differences between them and equations? Do inequalities have special properties? The goal of such questions is to: Understand what students are thinking. Elicit several answers from the students.
Properties of Inequality Let a, b and c represent real numbers. 1. If a < b, then a ± c < b ± c. 2. If a < b and if c > 0, then ac < bc. 3. If a < b and if c < 0, then ac > bc. 4.If a < b and if c > 0, then a/c < b/c. 5. If a < b and if c < 0, then a/c > b/c. Replacing < with >, ≤ , or ≥ results in similar properties.
Can we classify inequalities? According to what? Encourage students to jot down answers, then combine answers in a small group.
Linear Inequality in One Variable A linear inequality in one variable is an inequality that can be written in the form where a and b are real numbers, with a ≠ 0. (Any of the symbols ≥, <, and ≤ may also be used.) 0,ax b+ >
As we did with linear equations, we are looking for the solution set of the linear inequality. Invite students to talk about linear eq., other types of eq’s., and refresh their memories about sets as well as the concept of sol. set.
Example SOLVING A LINEAR INEQUALITY Solve Solution 3 5 7.x− + > − 3 5 7x− + > − 53 7 55x− > −− −+ Subtract 5. 3 12x− > − 3 12 3 3 x < − − − − Divide by –3. Reverse the direction of the inequality symbol when multiplying or dividing by a negative number. Don’t forget to reverse the symbol here. < = −∞4 ( ,4)x Combine like terms. Ask them, why is this linear?
Here (and after any major point) one can pause and show one or two multiple-choice questions, call for a vote (turn this into a competition or a debate). Solve A. B. C. D. 2 12 4 5 x x − ≥ − 10 3 x ≤ 30 11 x ≤ 10 3 x ≥ 30 11 x ≥
Application FINDING THE BREAK-EVEN POINT If the revenue and cost of a certain product are given by 4R x= and 2 1000,C x= + where x is the number of units produced and sold, at what production level does R at least equal C ? Let them try and guide them to the right thinking.
Implementing applications is attracting students and showing the beauty of mathematics. Let alone its crucial role in satisfying Bloom’s taxonomy
Application FINDING THE BREAK-EVEN POINT Solution For R at least equal C , this means R ≥ C . 4 2 1000x x≥ + Substitute. 2 1000x ≥ Subtract 2x. 500x ≥ Divide by 2. The break-even point (all costs that must be paid are paid, and there is neither profit nor loss) is at x = 500. This product will at least break even if the number of units produced and sold is in the interval [500, ∞).
When finishing some part of the lecture, allow time for questions. Doing so helps students to sort out the information. Then ask a question that motivates the next part. What if the inequality isn’t linear? What if we are dealing with quadratic inequality? What does quadratic inequality look like? How can we solve it? This kind of questions motivates students and plays a fundamental role in brainstorming.
Quadratic Inequalities A quadratic inequality is an inequality that can be written in the form 2 0ax bx c+ + < for real numbers a, b, and c, with a ≠ 0. (The symbol < can be replaced with >, ≤, or ≥.)
Solving a Quadratic Inequality Step 1 Solve the corresponding quadratic equation. Step 2 Identify the intervals determined by the solutions of the equation. Step 3 Use a test value from each interval to determine which intervals form the solution set.
Example SOLVING A QUADRATIC INEQULITY Solve Solution 2 12 0.x x− − < Step 1 Find the values of x that satisfy x2 – x – 12 = 0. ( 3)( 4) 0x x+ − = Factor. 3 0x + = 4 0x − =or Zero-factor property 3x = − 4x =or Solve each equation. Ask students: How can we solve this equation?
Example SOLVING A QUADRATIC INEQULITY Step 2 –3 0 4 Interval A Interval B Interval C (–∞, – 3) (–3, 4) (4, ∞)
Example SOLVING A QUADRATIC INEQULITY Interval Test Value Is x2 –x – 12 < 0 True or False? A: (–∞, –3) –4 (–4)2 – (–4) – 12 < 0 ? 8 < 0 False B: (–3, 4) 0 02 – 0 – 12 < 0 ? –12 < 0 True C: (4, ∞) 5 52 – 5 – 12 < 0 ? 8 < 0 False Step 3 Choose a test value from each interval. Since the values in Interval B make the inequality true, the solution set is (–3, 4).
Group work SOLVING A QUADRATIC INEQUALITY Solve the inequality Solution 2 2 5 12 0.x x+ − ≥ Step 1 Find the values of x that satisfy Corresponding quadratic equation (2 3)( 4) 0x x− + = Factor. 2 3 0x − = or 4 0x + = Zero-factor property 2 2 5 12 0.x x+ − = Divide students into small groups and let them solve together. This will indeed develop the students’ ability to communicate and work in teams. After they finish let students choose one of them to go to the board to show their work. •
Group work SOLVING A QUADRATIC INEQUALITY Solve Solution 2 2 5 12 0.x x+ − ≥ Step 1 2 3 0x − = or 4 0x + = 3 2 x = or 4x = −
Group work SOLVING A QUADRATIC INEQUALITY Solve Solution 2 2 5 12 0.x x+ − ≥ Step 2 The values form the intervals on the number line. –4 0 3/2 Interval A Interval B Interval C (–∞, – 4] [– 4, 3/2] [3/2, ∞)
Group work SOLVING A QUADRATIC INEQUALITY Step3 Choose a test value from each interval. Interval Test Value Is 2x2 + 5x – 12 ≥ 0 True or False? A: (–∞, –4] –5 2(–5)2 +5(–5) – 12 ≥ 0 ? 13 ≥ 0 True B: [– 4, 3/2] 0 2(0)2 +5(0) – 12 ≥ 0 ? –12 ≥ 0 False C: [3/2, ∞) 2 2(2)2 + 5(2) – 12 ≥ 0 ? 6 ≥ 0 True The values in Intervals A and C make the inequality true, so the solution set is the union of the intervals   −∞ − ∪ ∞÷  3 ( , 4] , . 2
Application FINDING PROJECTILE HEIGHT If a projectile is launched from ground level with an initial velocity of 96 ft per sec, its height s (in feet) t seconds after launching is given by the following equation. 2 16 96s t t= − + When will the projectile be greater than 80 ft above ground level? Here one should invite students to give some thoughts. Let them participate.
Application FINDING PROJECTILE HEIGHT 2 16 96 80t t− + > Set s greater than 80. Solution 2 16 96 80 0t t− + − > Subtract 80. 2 6 5 0t t + <− Divide by – 16. Reverse the direction of the inequality symbol. Now solve the corresponding equation.
Application FINDING PROJECTILE HEIGHT 2 6 5 0t t− + = Solution ( 1)( 5) 0t t− − = Factor. 1t = Zero-factor property or 5t = Solve each equation. or1 0t − = 5 0t − =
Application FINDING PROJECTILE HEIGHT Solution Interval C 1 0 5 Interval A Interval B (–∞, 1) (1, 5) (5, ∞) By testing the intervals, we observe that values in Interval B, (1, 5) , satisfy the inequality. The projectile is greater than 80 ft above ground level between 1 and 5 sec after it is launched.
Solving a Rational Inequality Step 1 Rewrite the inequality, if necessary, so that 0 is on one side and there is a single fraction on the other side. Step 2 Determine the values that will cause either the numerator or the denominator of the rational expression to equal 0. These values determine the intervals of the number line to consider.
Solving a Rational Inequality Step 3 Use a test value from each interval to determine which intervals form the solution set. A value causing the denominator to equal zero will never be included in the solution set. If the inequality is strict, any value causing the numerator to equal zero will be excluded. If the inequality is nonstrict, any such value will be included.
Can you apply these steps?
Try individually Solve Solution 5 1. 4x ≥ + Step 1 5 1 0 4x − ≥ + Subtract 1 so that 0 is on one side. ( )5 4 0 4 x x − + ≥ + Write as a single fraction. SOLVING A RATIONAL INEQUALITY Solving individually improves the ability to concentrate easier and work faster and helps in getting the whole credit for the work. (Some students prefer this) 1 4 0 x x − + ≥ Combine terms in the numerator, being careful with signs.
Example Solve Solution 5 1. 4x ≥ + Step 2 The quotient possibly changes sign only where x-values make the numerator or denominator 0. This occurs at 1 0x− = or 4 0x + = 1x = or 4x = − SOLVING A RATIONAL INEQUALITY Try individually
Example Solve Solution 5 1. 4x ≥ + Step 2 –4 0 1 Interval A Interval B Interval C (–∞, – 4) (–4, 1] [1, ∞) SOLVING A RATIONAL INEQUALITY Try individually
HOMEWORK 4 Step 3 Choose test values. Interval Test Value A: (–∞, –4) – 5 B: (–4, 1] 0 C: [1, ∞) 2 5 Is 1 True or False? 4x ≥ + ? 5 1 5 4 ≥ − + 5 1 False− ≥ 0 ? 5 1 4 ≥ + 5 1 e 4 Tru≥ 2 ? 5 1 4 ≥ + 5 1 e 6 Fals≥ SOLVING A RATIONAL INEQUALITY Try individually
Example 8 Step 3 The values in the interval (–4, 1) satisfy the original inequality. The value 1 makes the nonstrict inequality true, so it must be included in the solution set. Since –4 makes the denominator 0, it must be excluded. The solution set is (–4, 1]. SOLVING A RATIONAL INEQUALITYTry individually
Today’s lesson has finished Your leaving ticket is to sum up Ask students to say one or two sentences about the lesson before they leave.
Thanks fo r listening