What you should learn

To solve inequalities by using multiplication and division

NCTM Curriculm Standards 2, 4, 6 - 10

 

Introduction: A lever can be used to multiply the effort force you exert when trying to move something. The fixed point or fulcrum of a lever separates the length of the lever into two sections - the effort arm on which the effort force is applied and the resistance arm that exerts the resistance force. The mechanical advantage of a lever is the number of times a lever multiplies that effort force.

The formula for determining the mechanical advantage MA of a lever can be expressed as MA = Le/Lr, where Le represents the length of the effort arma dn Lr represents the length of the resistance arm.

Suppose a group of volunteers is clearing hiking trails at Yosemite National Park. They need to position a lever so that a mechanical advantage of at least 7 is achieved in order to remove a boulder blocking the trail. The volunteers place the lever on a rock sot hey can use the rockas a fulcrum. Theywill need to resistance arm to be 1.5 ft long so that it is long enought to get under the boulder. What should be the length of the lever in order to move the boulder?

We need to find the lenght of the effort arm to find the total length. Let Le represent the length of the effort arm. We know that 1/5 feet is the length of the resistance arm Lr. Since the mechanical advantage must be at least 7, we can write an inequality using the formula.

MA7

Le/Lr7

Le/1.57

If you were solving the equation Le/1.5 = 7, you would multiply each side by 1.5.

Will this method work when solving inequalities? Before answering this question, let's explore how multiplying (or dividing) an inequality by a posoitve or negative number affects the inequality. Conside the inequality 10 < 15, which we know is true.

Multiply by 2

10 < 15

10(2) < 15(2)

20 < 30 true

Multiply by -2

10 < 15

10(-2) < 15(-2)

-20 < -30 FALSE

-20 > -30 true

Divide by 5

10 < 15

10/5 < 15/5

2 < 3 true

Divide by -5

10 < 15

10/(-5) < 15/ (-5)

-2 < -3 FALSE

-2 > -3 true

These results suggest the following.

If each side of a true inequality is multiplied or divided by the same positive number, the resulting inequality is also true.

If each side of a true inequality is multiplied or divided by the same negative number, the direction of the inequality symbol must be reversed so that the resulting inequality is also true.

 

Multiplication and Division Properties for Inequalities: For all numbers, a, b, and c, the following are true

1. If c is positive and a < b, then ac < bc and a/c < b/c, and if c is positive and a > b, then ac > bc and a/c > b/c

2. If c is negative and a < b, then ac > bc and a/c > b/c, and if c is negative and a > b, then ac < bc and a/c < b/c

Exercise 1: Refer to the connection in the beginning of the lesson. What should the minimum length of the lever be?

Le/1.57

1.5 (Le/1.5)7 * 1.5

Le10.5

The effort arm must be at least 10.5 feet long.

In order to find the length of the lever, add the lengths of the effort arm and the resistance arm. The lever should be at least 10.5 + 1.5 or 12 feet long.

Exercise 2: Solve x/123/2

The solution set should be {x|x18}

Since dividing is the same as multiplying by the reciprocal, there can be two methods to solve an inequality that involves multiplication.

Exercise 3: Solve -3w > 27

Exercise 4: Angelica Moreno is a sales representative for an appliance distributor. She needs at least $5000 in weekly sales of a particular TV model to qualify for a sales copetition to win a trip to the Bahamas. If the TVs sell for $250 each, how many TVs will Ms. Moreno have to sell to qualify?

Explore

Let t represent the number of TVs to be sold. At least $5000 means greater than or equal to $5000

Plan

The price of one TV times the number of sets sold must be greater than or equal to the total amount of sales needed.

$250 * t$5000

Solve Continue from here...

Exercise 5: Triangle XYZ is not an acute triangle. The greatest angle in the triangle has a measure of (6d) degrees. What are the possible values of d?

SinceXYZ is not acute, the measure of the greatest angle must be 90 degrees or larger, but less than 180 degrees. Thus, 6d90 and 6d < 180. (Take it from here...)

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 21 - 49 odd, 50, 51, 53 - 62

Alternative Homework: Enriched: 20 - 48 even, 50 - 63

Extra Practice: Students book page 772 Lesson 7-2

Extra Practice Worksheet: Click Here.

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