2.5 Solve Linear Inequalities

Learning objectives

Learning objectives

• Graph inequalities on the number line

• Solve linear inequalities

• Translate words to an inequality and solve

• Solve applications with linear inequalities

Graph Inequalities on the Number Line

What number would make the inequality $x>3$ true? Are you thinking, “x could be four”? That’s correct, but x could be 6, too, or 37, or even 3.001. Any number greater than three is a solution to the inequality $x>3.$

We show all the solutions to the inequality $x>3$ on the number line by shading in all the numbers to the right of three, to show that all numbers greater than three are solutions. Because the number three itself is not a solution, we put an open parenthesis at three.

We can also represent inequalities using interval notation. There is no upper end to the solution to this inequality. In interval notation, we express $x>3$ as $(3,\infty).$ The symbol $\infty$ is read as “infinity.” It is not an actual number.

Figure 2.2 shows both the number line and the interval notation.

We use the left parenthesis symbol, (, to show that the endpoint of the inequality is not included. The left bracket symbol, [, shows that the endpoint is included.

The inequality $x\le{}1$ means all numbers less than or equal to one. Here we need to show that one is a solution, too. We do that by putting a bracket at $x=1.$ We then shade in all the numbers to the left of one, to show that all numbers less than one are solutions. See Figure 2.3.

There is no lower end to those numbers. We write $x\le{}1$ in interval notation as $(-\infty,1].$ The symbol $-\infty$ is read as “negative infinity.” Figure 2.3 shows both the number line and interval notation.

The notation for inequalities on a number line and in interval notation use the same symbols to express the endpoints of intervals.

Example 2.48

Solution

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What numbers are greater than two but less than five? Are you thinking say, $2.5,3,3\frac{2}{3},4,4.99$? We can represent all the numbers between two and five with the inequality $2<x<5.$ We can show $2<x<5$ on the number line by shading all the numbers between two and five. Again, we use the parentheses to show the numbers two and five are not included. See Figure 2.4.

Example 2.49

Solution

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Solve Linear Inequalities

A linear inequality is much like a linear equation—but the equal sign is replaced with an inequality sign. A linear inequality is an inequality in one variable that can be written in one of the forms, $ax+b<c,$$ax+b\le{}c,$$ax+b>c,$ or $ax+b\ge{}c.$

When we solved linear equations, we were able to use the properties of equality to add, subtract, multiply, or divide both sides and still keep the equality. Similar properties hold true for inequalities.

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality. For example:

Notice that the inequality sign stayed the same.

This leads us to the Addition and Subtraction Properties of Inequality.

What happens to an inequality when we divide or multiply both sides by a constant?

Let’s first multiply and divide both sides by a positive number.

The inequality signs stayed the same.

Does the inequality stay the same when we divide or multiply by a negative number?

Notice that when we filled in the inequality signs, the inequality signs reversed their direction.

When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.

This gives us the Multiplication and Division Property of Inequality.

When we divide or multiply an inequality by a:

• positive number, the inequality stays the same.

• negative number, the inequality reverses.

Sometimes when solving an inequality, as in the next example, the variable ends upon the right. We can rewrite the inequality in reverse to get the variable to the left.

Think about it as “If Xander is taller than Andy, then Andy is shorter than Xander.”

Example 2.50

Solution

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Be careful when you multiply or divide by a negative number—remember to reverse the inequality sign.

Example 2.51

Solution

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Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but make sure to pay close attention when we multiply or divide to isolate the variable.

Example 2.52

Solution

When solving inequalities, it is usually easiest to collect the variables on the side where the coefficient of the variable is largest. This eliminates negative coefficients and so we don’t have to multiply or divide by a negative—which means we don’t have to remember to reverse the inequality sign.

Example 2.53

Solution

Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.

Example 2.54

Solution

We can clear fractions in inequalities much as we did in equations. Again, be careful with the signs when multiplying or dividing by a negative.

Example 2.55

Solution

Translate to an Inequality and Solve

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like “more than” and “less than.” But others are not as obvious. Table 2.2 shows some common phrases that indicate inequalities.

$>$ · $\ge$ · $<$ · $\le$ is greater thanis more thanis larger thanexceeds · is greater than or equal tois at leastis no less thanis the minimum · is less thanis smaller thanhas fewer thanis lower than · is less than or equal tois at mostis no more thanis the maximum

Table

2.2

Example 2.56

Solution

Solve Applications with Linear Inequalities

Many real-life situations require us to solve inequalities. The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations.

We will read the problem and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality.

Sometimes an application requires the solution to be a whole number, but the algebraic solution to the inequality is not a whole number. In that case, we must round the algebraic solution to a whole number. The context of the application will determine whether we round up or down.

Example 2.57

Solution

Step 1. Read the problem.

Step 2. Identify what you are looking for. · the maximum number of tablets Dawn can buy Step 3. Name what you are looking for.

Choose a variable to represent that quantity. · $\mathrm{Let}\,n=\mathrm{the number of tablets.}$ Step 4. Translate. Write a sentence that gives the information to find it. · $254.12 times the number of tablets is no more than$4,000.

Translate into an inequality. · $254.12n\le{}4000$ Step 5. Solve the inequality.But n must be a whole number of tablets, so round to 15. · $\begin{array}{l} \,n\le{}15.74 \\ \,n\le{}15 \\ \end{array}$ Step 6. Check the answer in the problem and make sure it makes sense.

Rounding down the price to $250, 15 tablets would cost$3,750, while 16 tablets would be $4,000. So a maximum of 15 tablets at$254.12 seems reasonable.

Step 7. Answer the question with a complete sentence. · Dawn can buy a maximum of 15 tablets.

Example 2.58

Solution

Step 1. Read the problem.

Step 2. Identify what you are looking for. · the number of text messages Taleisha can make Step 3. Name what you are looking for.

Choose a variable to represent that quantity. · $\mathrm{Let}\,t=\mathrm{the number of text messages.}$ Step 4. Translate Write a sentence that gives the information to find it. · $28.80 plus$0.20 times the number of text messages is less than or equal to $50.

Translate into an inequality. · $\,28.80+0.20t\le{}50$ Step 5. Solve the inequality. · $\begin{array}{l} \,0.2t\le{}21.2 \\ \,t\le{}106\,\mathrm{text messages} \\ \end{array}$ Step 6. Check the answer in the problem and make sure it makes sense.$\mathrm{Yes,}\,28.80+0.20(106)=50.$ Step 7. Write a sentence that answers the question. · Taleisha can send/receive no more than 106 text messages to keep her bill no more than $50.

Profit is the money that remains when the costs have been subtracted from the revenue. In the next example, we will find the number of jobs a small businesswoman needs to do every month in order to make a certain amount of profit.

Example 2.59

Solution

Step 1. Read the problem.

Step 2. Identify what you are looking for. · the number of invitations Felicity needs to write Step 3. Name what you are looking for.Choose a variable to represent it. · $\mathrm{Let}\,j=\mathrm{the number of invitations.}$ Step 4. Translate. Write a sentence that gives the information to find it. · $2.50 times the number of invitations minus$650 is at least $2,800.

Translate into an inequality. · $2.50j-650\ge{}2,800$ Step 5. Solve the inequality. · $\begin{array}{l} \,2.5j\ge{}3,450 \\ \,j\ge{}1,380\,\mathrm{invitations} \\ \end{array}$ Step 6. Check the answer in the problem and make sure it makes sense.

If Felicity wrote 1400 invitations, her profit would be2.50(1400) − 650, or $2,850. This is more than$2800.

Step 7. Write a sentence that answers the question. · Felicity must write at least 1,380 invitations.

There are many situations in which several quantities contribute to the total expense. We must make sure to account for all the individual expenses when we solve problems like this.

Example 2.60

Solution

Step 1. Read the problem.

Step 2. Identify what you are looking for. · the number of hours Malik must tutor Step 3. Name what you are looking for.

Choose a variable to represent that quantity. · $\mathrm{Let}\,h=\mathrm{the number of hours.}$ Step 4. Translate. Write a sentence that gives the information to find it. · The expenses must be less than or equal to the income. The cost of airfare plus the cost of food and sightseeing and the hotel bill must be less than the savings plus the amount earned tutoring.

Translate into an inequality. · $525+780+95(6)\le{}840+45h$ Step 5. Solve the inequality. · $\,\begin{array}{l} 1,875\le{}840+45h \\ 1,035\le{}45h \\ \,23\le{}h \\ \,h\ge{}23 \\ \end{array}$ Step 6. Check the answer in the problem and make sure it makes sense.We substitute 23 into the inequality.$\begin{array}{l} \,1,875\le{}840+45h \\ \,1,875\le{}840+45(23) \\ \,1,875\le{}1875 \\ \end{array}$ Step 7. Write a sentence that answers the question. · Malik must tutor at least 23 hours.

Section 2.5 Exercises

Section 2.5 Exercises

Graph Inequalities on the Number Line

In the following exercises, graph each inequality on the number line and write in interval notation.

Solve Linear Inequalities

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

Translate to an Inequality and Solve

In the following exercises, translate and solve. Then graph the solution on the number line and write the solution in interval notation.

Solve Applications with Linear Inequalities

In the following exercises, solve.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

1. Graph each inequality on the number line and write in interval notation: ⓐ $x>2$ ⓑ $x\le-1.5$ ⓒ $x\ge\frac{3}{4}.$

2. Graph each inequality on the number line and write in interval notation: ⓐ $x\le-4$ ⓑ $x\ge{}0.5$ ⓒ $x<-\frac{2}{3}.$

3. Graph each inequality on the number line and write in interval notation:

ⓐ $-2<x<1$ ⓑ $-5\le{}x<-4$ ⓒ $1\le{}x\le{}4.25$

4. Graph each inequality on the number line and write in interval notation:

ⓐ $-6<x<2$ ⓑ $-3\le{}x<-1$ ⓒ $2.5\le{}x\le{}6$

1. Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

ⓐ $p-\frac{3}{4}\ge\frac{1}{6}$ ⓑ $9c>72$ ⓒ $24\le\frac{3}{8}m$

2. Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

ⓐ $r-\frac{1}{3}\le\frac{7}{12}$ ⓑ $12d\le{}60$ ⓒ $-24<\frac{4}{3}n$

3. Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

ⓐ $-8q<32$ ⓑ $\frac{k}{-12}\le{}15.$

4. Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

ⓐ $-7r\le-70$ ⓑ $\frac{u}{-4}\ge-16.$

1. Solve the inequality, graph the solution on the number line, and write the solution in interval notation: $3q\ge{}7q-23.$

2. Solve the inequality, graph the solution on the number line, and write the solution in interval notation: $6x<10x+19.$

3. Solve the inequality $9y+2(y+6)>5y-24$, graph the solution on the number line, and write the solution in interval notation.

4. Solve the inequality $6u+8(u-1)>10u+32$, graph the solution on the number line, and write the solution in interval notation.

1. Solve the inequality $4b-3(3-b)>5(b-6)+2b$, graph the solution on the number line, and write the solution in interval notation.

2. Solve the inequality $9h-7(2-h)<8(h+11)+8h$, graph the solution on the number line, and write the solution in interval notation.

3. Solve the inequality $\frac{1}{4}x-\frac{1}{12}x>\frac{1}{6}x+\frac{7}{8}$, graph the solution on the number line, and write the solution in interval notation.

4. Solve the inequality $\frac{2}{5}z-\frac{1}{3}z<\frac{1}{15}z+\frac{3}{5}$, graph the solution on the number line, and write the solution in interval notation.

1. Sergio and Lizeth have a very tight vacation budget. They plan to rent a car from a company that charges $75 a week plus$0.25 a mile. How many miles can they travel during the week and still keep within their $200 budget?

2. Rameen’s heating bill is $5.42 per month plus$1.08 per therm. How many therms can Rameen use if he wants his heating bill to be a maximum of $87.50.

3. Caleb has a pet sitting business. He charges $32 per hour. His monthly expenses are$2,272. How many hours must he work in order to earn a profit of at least $800 per month?

4. Elliot has a landscape maintenance business. His monthly expenses are $1,100. If he charges$60 per job, how many jobs must he do to earn a profit of at least $4,000 a month?

1. Brenda’s best friend is having a destination wedding and the event will require 3 nights in a hotel. Brenda has $500 in savings and can earn$15 an hour babysitting. She expects to pay $350 airfare,$375 for food and entertainment and $60 a night for her share of a hotel room. How many hours must she babysit to have enough money to pay for the trip?

2. Josue wants to go on a 10-night road trip with friends next spring. It will cost him $180 for gas,$450 for food, and $49 per night to share a motel room. He has$520 in savings and can earn $30 per driveway shoveling snow. How many driveways must he shovel to have enough money to pay for the trip?

3. ⓐ $x>3$ⓑ $x\le-0.5$ⓒ $x\ge\frac{1}{3}$

4. ⓐ $x\le{}5$ⓑ $x\ge-1.5$ⓒ $x<-\frac{7}{3}$

1. ⓐ $-2<x<0$ⓑ $-5\le{}x<-3$ⓒ $0\le{}x\le{}3.5$

2. ⓐ $-4<x<2$ⓑ $-5<x\le-2$ⓒ $-3.75\le{}x\le{}0$

3. ⓐ $b+\frac{7}{8}\ge\frac{1}{6}$ⓑ $6y<48$ⓒ $40<\frac{5}{8}k$

4. ⓐ $g-\frac{11}{12}<-\frac{5}{18}$ⓑ $7s<-28$ⓒ $\frac{9}{4}g\le{}36$

1. ⓐ $-8v\le{}96$ⓑ $\frac{b}{-10}\ge{}30$

2. ⓐ $-7d>105$ⓑ $-18>\frac{q}{-6}$

3. The elevator in Yehire’s apartment building has a sign that says the maximum weight is 2100 pounds. If the average weight of one person is 150 pounds, how many people can safely ride the elevator?

4. Arleen got a $20 gift card for the coffee shop. Her favorite iced drink costs$3.79. What is the maximum number of drinks she can buy with the gift card?

1. Ryan charges his neighbors $17.50 to wash their car. How many cars must he wash next summer if his goal is to earn at least$1,500?

2. Kimuyen needs to earn $4,150 per month in order to pay all her expenses. Her job pays her$3,475 per month plus 4% of her total sales. What is the minimum Kimuyen’s total sales must be in order for her to pay all her expenses?

3. Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her$66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?

4. Kiyoshi’s phone plan costs $17.50 per month plus$0.15 per text message. What is the maximum number of text messages Kiyoshi can use so the phone bill is no more than $56.60?

1. Kellen wants to rent a banquet room in a restaurant for her cousin’s baby shower. The restaurant charges $350 for the banquet room plus$32.50 per person for lunch. How many people can Kellen have at the shower if she wants the maximum cost to be $1,500?

2. Noe installs and configures software on home computers. He charges $125 per job. His monthly expenses are$1,600. How many jobs must he work in order to make a profit of at least $2,400?

3. Melissa makes necklaces and sells them online. She charges $88 per necklace. Her monthly expenses are$3,745. How many necklaces must she sell if she wants to make a profit of at least $1,650?

4. Cesar is planning a four-day trip to visit his friend at a college in another state. It will cost him $198 for airfare,$56 for local transportation, and $45 per day for food. He has$189 in savings and can earn $35 for each lawn he mows. How many lawns must he mow to have enough money to pay for the trip?

1. Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.

Nineteen less than p is no less than 47.

2. Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.

Four more than a is at most 15.

3. Angie has $20 to spend on juice boxes for her son’s preschool picnic. Each pack of juice boxes costs$2.63. What is the maximum number of packs she can buy?

4. Daniel wants to surprise his girlfriend with a birthday party at her favorite restaurant. It will cost $42.75 per person for dinner, including tip and tax. His budget for the party is$500. What is the maximum number of people Daniel can have at the party?

1. Eun-Kyung works as a tutor and earns $60 per hour. She has$792 in savings. She is planning an anniversary party for her parents. She would like to invite 40 guests. The party will cost her $1,520 for food and drinks and$150 for the photographer. She will also have a favor for each of the guests, and each favor will cost $7.50. How many hours must she tutor to have enough money for the party?

2. Maximum weight on a boat In 2004, a water taxi sank in Baltimore harbor and five people drowned. The water taxi had a maximum capacity of 3,500 pounds (25 people with average weight 140 pounds). The average weight of the 25 people on the water taxi when it sank was 168 pounds per person. What should the maximum number of people of this weight have been?

3. Shower budget Penny is planning a baby shower for her daughter-in-law. The restaurant charges $950 for up to 25 guests, plus$31.95 for each additional guest. How many guests can attend if Penny wants the total cost to be no more than $1,500?

4. Explain why it is necessary to reverse the inequality when solving $\frac{n}{-3}<12.$

1. Find out how many units you have left, after this term, to achieve your college goal and estimate the number of units you can take each term in college. Calculate the number of terms it will take you to achieve your college goal by writing an appropriate inequality and then solving it. Is this an acceptable number of terms until you meet your goal? What are some ways you could accelerate this process?