Chapter 2 Linear Equations and Inequalities with One Variable

2.1 Addition and Subtraction Properties of Equality2.2 Multiplication and Division Properties of Equality2.3 Solving Equations with Variables on Both Sides2.4 Solving and Graphing Linear Inequalities on a Number Line

Identify each of the following as an expression or an equation. Do not solve or simplify.a. 5x + 7

b. 8x + 7 =12

c. 24 = 24

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Determine whether or not the given values of the variables are solutions to the given equation.a. C = 10.50h, where C is the cost for a babysitter in dollars (output) for h hours (input); h = 4 and C = 42. Explain what these values mean in terms of this situation.

b. T = 4c where T is the time in minutes it takes to produce c toy cars; c = 35 and T = 130. Explain what these values mean in terms of this situation.

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Determine whether or not the given values of the variables are solutions to the given equation.

c. 5x + 18 = 8x; x = 6

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A person’s total calorie intake is made up of the calories from meals, snacks, and drinks. The total calorie intake can be represented by the equation T = m + s + d, where T is the total calorie intake, m is the calories eaten during meals, s is the calories eaten during snacks, and d is calories taken in through drinks. Use this equation to answer the following questions.a.A man who needs 2500 calories a day to maintain his weight drinks 300 calories worth of soda a day and has 400 calories worth of snacks. How many calories can he eat during meals to maintain his weight?

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ts

A person’s total calorie intake is made up of the calories from meals, snacks, and drinks. The total calorie intake can be represented by the equation T = m + s + d, where T is the total calorie intake, m is the calories eaten during meals, s is the calories eaten during snacks, and d is calories taken in through drinks. Use this equation to answer the following questions.b.A woman who needs 2000 calories a day to maintain her weight wants to go out for a late snack of ice cream tonight. If she drinks no calories that day and wants 540 calories (two scoops) of ice cream tonight, how many total calories can she eat at her meals that day?

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Solve the equation. Check the answer.a. x + 35 = 40 b. g – 14 = 11

c. 2.73 + m = 3.85

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Use the equation P = R – C, where P is the profit in dollars, R is the revenue in dollars, and C is the cost in dollars for the company.a. If a company has a monthly revenue of $120,000 and monthly costs of $98,000, what is its profit?

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Use the equation P = R – C, where P is the profit in dollars, R is the revenue in dollars, and C is the cost in dollars for the company.

b. If a company wants a profit of $40,000 for the month and has costs of $60,000, what revenue will the company have to generate?

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Solve the equation. Check the answer.

a. b.

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2 5

7 7x

3 7

4 12m

Solve the equation. Check the answer.a. 12 + b = 38 b. 17 = 27 + x

c. –11 + m = –17

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Solve the following literal equations for the indicated variable.a. The perimeter of an isosceles trapezoid (see image below)

P = 2a + B + b for B

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Solve the following literal equations for the indicated variable.b. C = rm + f ; for f c. G = w – b ; for w

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Melinda works at a local coffee shop and earns $11 per hour after taxes are taken out of her paycheck. Melinda can calculate her weekly take-home pay using the equation P = 11h, where P represents Melinda’s take-home pay in dollars when she works h hours in a week.a. Calculate Melinda’s take-home pay if she works 24 hours a week.

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Melinda works at a local coffee shop and earns $11 per hour after taxes are taken out of her paycheck. Melinda can calculate her weekly take-home pay using the equation P = 11h, where P represents Melinda’s take-home pay in dollars when she works h hours in a week.b. How many hours does Melinda need to work in a week if she needs $192.50 to pay her car insurance?

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Solve the equation. Check the answer.a. 4n = 48 b. –6w = 81

c. –a = 24

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Solve the equation. Check the answer.

a. b.

87

x

212

5h

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The population of Nebraska can be estimated by using the equation P = 9t + 1706, where P is the population in thousands and t = the number of years since 2000.a. Estimate the population of Nebraska in 2010.

b. Find what year Nebraska will have a population of 1850 thousand people.

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Solve the equation. Check the answer.

a. 5t + 14 = 44 b.

c. –38 = 4g + 10 + 8g

18 23

k

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Translate each of the following sentences into an equation. Solve the equation. Check the answer.a. Three times a number added to 14 is equal to 35.

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Translate each of the following sentences into an equation. Solve the equation. Check the answer.b. Six times the difference of a number and 20 is 54.

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Translate each of the following sentences into an equation. Solve the equation. Check the answer.c. The perimeter of a triangle is 12 meters, and the lengths of two sides of the triangle are 4 meters and 6 meters. Find the length of the triangle’s third side.

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Define the variables and translate each sentence into an equation. Solve the equation. Check the answer for accuracy and reasonableness.a.The Be Tough Karate Studio charges a $100 fee to sign up for lessons and $20 per lesson taken. If Mark paid $2000 for his daughter’s karate lessons this year, how many lessons did she take?

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Define the variables and translate each sentence into an equation. Solve the equation. Check the answer for accuracy and reasonableness.b. The local YMCA charges a one-time $150 fee and $74 per month for a family health club membership. For how many months can a family prepay if they have $890?

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Solve each literal equation for the indicated variable.a. Pay: P = 11h; solve for h.

b. The perimeter of an isosceles trapezoidP = 2a + B + b for a

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Solve each equation for the indicated variable.a. 5x + 7y = 42; for y

b. 3

3 9 (4 8);for 4

r t t

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Solve each equation. Check the answer.a. 6x + 18 = 4x + 30 b. 5 (2p – 9) = 8p – 10 – 5p

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Solve each equation. Check the answer.c. 0.4 (6n – 8) + 0.3n = 6.2n – 21.4

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Solve each equation. Check the answer.a. b.

2.3-2

2 48

5 5y 7 37 3

34 14 4m m m

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Solve each equation. Check the answer.c.

2.3-2

3 7 37(5 6)

8 8 4t t

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Solve each equation. If the equation has no solution, write “no solution.” If it is an identity, write “all real numbers.”a.6x + 5 = 2(4x – 8) – 2x

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Solve each equation. If the equation has no solution, write “no solution.” If it is an identity, write “all real numbers.”b.

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27 5 2 (3 12) 3 3

3x x x x

Solve each equation. If the equation has no solution, write “no solution.” If it is an identity, write “all real numbers.”c. 9x + 2 = 2

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Translate each of the following sentences into an equation and solve.a. Eight less than four times a number is equal to six times the sum of the number and 5.

b. The quotient of a number and negative five is equal to twice the number minus thirty-three.

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Given the following triangle find the measure of eachAngle.

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Jim is leading a group of boys on an all-day hike and needs to buy food and supplies for the group. Jim wants to purchase three sports drinks, two protein bars, and a candy bar for each person on the hike. Jim also wants to bring six extra sports drinks and three extra protein bars. Sports drinks cost $0.75 each, protein bars cost $1.25 each, and candy bars cost $1 each. If Jim has $60 to buy supplies, how many people can go on the hike?

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In the first Classroom Example in Section 2.2, we considered Melinda’s weekly take-home pay at a local coffee shop. Melinda earned $11 per hour. Let h = the number of hours Melinda works in a week.a.Write an inequality to show that Melinda needs to earn at least $192.50 in take-home pay to pay her car insurance.

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In the first Classroom Example in Section 2.2, we considered Melinda’s weekly take-home pay at a local coffee shop. Melinda earned $11 per hour. Let h = the number of hours Melinda works in a week.b. Solve the inequality found in part a. Write the solution as a complete Sentence.

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In the first Classroom Example in Section 2.2, we considered Melinda’s weekly take-home pay at a local coffee shop. Melinda earned $11 per hour. Let h = the number of hours Melinda works in a week.c. Find the number of hours Melinda needs to work in a week to earn at least $352.

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Solve the following inequalities. Check the answer.a. 7x + 12 > 40

b. 7x +15 – 12x ≥ 75

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Solve the following inequalities. Check the answer.c.

d.8(p + 4) ≤ 12p – 36

18 67

m

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Rewrite each inequality so that the variable appears on the left side.a. 12 < x

b. –24 ≥ h

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A housecleaning service charges clients $45 for the first hour of cleaning and $30 for each additional hour. Let h be the number of additional hours after the first hour.a. Write an inequality to show the housecleaning service charging at most $150 (the client’s budget).

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A housecleaning service charges clients $45 for the first hour of cleaning and $30 for each additional hour. Let h be the number of additional hours after the first hour.b. Solve the inequality from part a. Write the solution in a complete sentence.

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Solve the following inequalities. Write the solution set using both interval notation and a number line.a. 4x < 20

b. 24

x

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Given the interval form of an inequality, graph thesolution set on a number line. Then rewrite using

inequality notation.a. (4, ∞)

b. (–∞, –2]

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Write a compound inequality for the following Statements.a. NFL teams are allowed no more than 45 active players on their rosters.

b. A local restaurant will reserve a back room for groups up to 40. The minimum size group allowed to reserve the room is 15.

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Write a compound inequality for the following Statements.c. It is commonly accepted that most wines should be stored at a temperature between 55°F and 60°F.

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Solve each compound inequality. Write the answer inthe requested form.a. 16 < 3x –2 < 25 Answer using a number line.

b. Answer using interval notation.

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