Solving Applications With Linear Inequalities

Emma got a new job and will have to move. Her monthly income will be $5,265. To qualify to rent an apartment, Emma’s monthly income must be at least three times as much as the rent. What is the highest rent Emma will qualify for?

Solution

\(\begin{array}{cccc}\mathbf{\text{Step 1. Read}}\phantom{\rule{0.2em}{0ex}}\text{the problem.}\hfill & & & \\ \mathbf{\text{Step 2. Identify}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill & & & \text{the highest rent Emma will qualify for}\hfill \\ \begin{array}{c}\mathbf{\text{Step 3. Name}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill \\ \phantom{\rule{2em}{0ex}}\text{Choose a variable to represent that quantity.}\hfill \end{array}\hfill & & & \text{Let}\phantom{\rule{0.2em}{0ex}}r=\text{the rent}.\hfill \\ \begin{array}{c}\mathbf{\text{Step 4. Translate}}\phantom{\rule{0.2em}{0ex}}\text{into an inequality.}\hfill \\ \phantom{\rule{2em}{0ex}}\text{First write a sentence that gives the information}\hfill \\ \phantom{\rule{2em}{0ex}}\text{to find it.}\hfill \end{array}\hfill & & & \begin{array}{c}\text{Emma’s monthly income must be at least}\hfill \\ \text{three times the rent.}\hfill \end{array}\hfill \\ \begin{array}{c}\mathbf{\text{Step 5. Solve}}\phantom{\rule{0.2em}{0ex}}\text{the inequality.}\hfill \\ \phantom{\rule{2em}{0ex}}\text{Remember,}\phantom{\rule{0.2em}{0ex}}a>x\phantom{\rule{0.2em}{0ex}}\text{has the same meaning}\hfill \\ \phantom{\rule{2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}x

Dawn won a mini-grant of $4,000 to buy tablet computers for her classroom. The tablets she would like to buy cost $254.12 each, including tax and delivery. What is the maximum number of tablets Dawn can buy?

Solution

\(\begin{array}{cccc}\mathbf{\text{Step 1. Read}}\phantom{\rule{0.2em}{0ex}}\text{the problem.}\hfill & & & \\ \mathbf{\text{Step 2. Identify}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill & & & \text{the maximum number of tablets Dawn can buy}\hfill \\ \begin{array}{c}\mathbf{\text{Step 3. Name}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill \\ \phantom{\rule{2em}{0ex}}\text{Choose a variable to represent that quantity.}\hfill \end{array}\hfill & & & \text{Let}\phantom{\rule{0.2em}{0ex}}n=\text{the number of tablets}.\hfill \\ \begin{array}{c}\mathbf{\text{Step 4. Translate}}.\phantom{\rule{0.2em}{0ex}}\text{Write a sentence that}\hfill \\ \text{gives the information to find it.}\hfill \\ \phantom{\rule{2em}{0ex}}\text{Translate into an inequality}.\hfill \end{array}\hfill & & & \begin{array}{c}\text{\$254.12 times the number of tablets is no}\hfill \\ \text{more than \$4,000.}\hfill \\ \hfill \phantom{\rule{0.5em}{0ex}}254.12n\le 4,000\hfill \end{array}\hfill \\ \mathbf{\text{Step 5. Solve}}\phantom{\rule{0.2em}{0ex}}\text{the inequality}.\hfill & & & \hfill \phantom{\rule{0.8em}{0ex}}n\le 15.74\hfill \\ \begin{array}{c}\phantom{\rule{2em}{0ex}}\text{But}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{must be a whole number of tablets,}\hfill \\ \phantom{\rule{3em}{0ex}}\text{so round to 15.}\hfill \end{array}\hfill & & & \hfill n\le 15\hfill \\ \begin{array}{c}\mathbf{\text{Step 6. Check}}\phantom{\rule{0.2em}{0ex}}\text{the answer in the problem}\hfill \\ \text{and make sure it makes sense.}\hfill \end{array}\hfill & & & \\ \begin{array}{}\\ \phantom{\rule{2.5em}{0ex}}\text{Rounding down the price to \$250,}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{15 tablets would cost \$3,750, while}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{16 tablets would be \$4,000. So a}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{maximum of 15 tablets at \$254.12}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{seems reasonable.}\hfill \end{array}\hfill & & & \\ \begin{array}{c}\mathbf{\text{Step 7. Answer}}\phantom{\rule{0.2em}{0ex}}\text{the question with a}\hfill \\ \text{complete sentence.}\hfill \end{array}\hfill & & & \text{Dawn can buy a maximum of 15 tablets.}\hfill \end{array}\)

Pete works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925?

Solution

\(\begin{array}{cccc}\mathbf{\text{Step 1. Read}}\phantom{\rule{0.2em}{0ex}}\text{the problem.}\hfill & & & \\ \mathbf{\text{Step 2. Identify}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill & & & \begin{array}{c}\text{the total sales needed for his variable pay}\hfill \\ \text{option to exceed the fixed amount of \$925}\hfill \end{array}\hfill \\ \begin{array}{c}\mathbf{\text{Step 3. Name}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill \\ \phantom{\rule{2em}{0ex}}\text{Choose a variable to represent that quantity.}\hfill \end{array}\hfill & & & \text{Let}\phantom{\rule{0.2em}{0ex}}s=\text{the total sales}.\hfill \\ \begin{array}{c}\mathbf{\text{Step 4. Translate}}\phantom{\rule{0.2em}{0ex}}\text{Write a sentence that gives}\hfill \\ \text{the information to find it.}\hfill \end{array}\hfill & & & \text{\$500 plus 12% of total sales is more than \$925.}\hfill \\ \begin{array}{c}\phantom{\rule{2em}{0ex}}\text{Translate into an inequality. Remember to}\hfill \\ \phantom{\rule{3em}{0ex}}\text{convert the percent to a decimal.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{0.1em}{0ex}}500+0.12s>925\hfill \\ \mathbf{\text{Step 5. Solve}}\phantom{\rule{0.2em}{0ex}}\text{the inequality.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\begin{array}{ccc}\hfill 0.12s& >\hfill & 425\hfill \\ \hfill s& >\hfill & 3,541.\stackrel{\text{—}}{66}\hfill \end{array}\hfill \\ \begin{array}{c}\mathbf{\text{Step 6. Check}}\phantom{\rule{0.2em}{0ex}}\text{the answer in the problem and}\hfill \\ \text{make sure it makes sense.}\hfill \end{array}\hfill & & & \\ \begin{array}{}\\ \phantom{\rule{2.5em}{0ex}}\text{If we round the total sales up to}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{\$4,000, we see that}\hfill \\ \phantom{\rule{2.5em}{0ex}}500+0.12\left(4,000\right)=980,\text{which is more}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{than \$925.}\hfill \end{array}\hfill & & & \\ \mathbf{\text{Step 7. Answer}}\phantom{\rule{0.2em}{0ex}}\text{the question with a complete sentence.}\hfill & & & \text{The total sales must be more than \$3,541.67.}\hfill \end{array}\)

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 and still keep within their $200 budget?

Solution

\(\begin{array}{cccc}\mathbf{\text{Step 1. Read}}\phantom{\rule{0.2em}{0ex}}\text{the problem.}\hfill & & & \\ \mathbf{\text{Step 2. Identify}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill & & & \text{the number of miles Sergio and Lizeth can travel}\hfill \\ \begin{array}{c}\mathbf{\text{Step 3. Name}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill \\ \phantom{\rule{2em}{0ex}}\text{Choose a variable to represent that quantity.}\hfill \end{array}\hfill & & & \text{Let}\phantom{\rule{0.2em}{0ex}}m=\text{the number of miles}.\hfill \\ \begin{array}{c}\mathbf{\text{Step 4. Translate}}\phantom{\rule{0.2em}{0ex}}\text{Write a sentence that}\hfill \\ \text{gives the information to find it.}\hfill \end{array}\hfill & & & \begin{array}{c}\text{\$75 plus 0.25 times the number of miles is}\hfill \\ \text{less than or equal to \$200.}\hfill \end{array}\hfill \\ \phantom{\rule{2em}{0ex}}\text{Translate into an inequality.}\hfill & & & \hfill 75+0.25m\le 200\hfill \\ \mathbf{\text{Step 5. Solve}}\phantom{\rule{0.2em}{0ex}}\text{the inequality.}\hfill & & & \hfill \phantom{\rule{4.5em}{0ex}}\begin{array}{ccc}\hfill 0.25m& \le \hfill & 125\hfill \\ \hfill m& \le \hfill & 500\phantom{\rule{0.2em}{0ex}}\text{miles}\hfill \end{array}\hfill \\ \begin{array}{c}\mathbf{\text{Step 6. Check}}\phantom{\rule{0.2em}{0ex}}\text{the answer in the problem}\hfill \\ \text{and make sure it makes sense.}\hfill \end{array}\hfill & & & \\ \phantom{\rule{2.5em}{0ex}}\text{Yes,}\phantom{\rule{0.2em}{0ex}}75+0.25\left(500\right)=200.\hfill & & & \\ \mathbf{\text{Step 7. Write}}\phantom{\rule{0.2em}{0ex}}\text{a sentence that answers the question.}\hfill & & & \begin{array}{c}\text{Sergio and Lizeth can travel 500 miles}\hfill \\ \text{and still stay on budget.}\hfill \end{array}\hfill \end{array}\)

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?

Solution

\(\begin{array}{cccc}\mathbf{\text{Step 1. Read}}\phantom{\rule{0.2em}{0ex}}\text{the problem.}\hfill & & & \\ \mathbf{\text{Step 2. Identify}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill & & & \text{the number of jobs Elliot needs}\hfill \\ \begin{array}{c}\mathbf{\text{Step 3. Name}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for. Choose}\hfill \\ \text{a variable to represent it.}\hfill \end{array}\hfill & & & \text{Let}\phantom{\rule{0.2em}{0ex}}j=\text{the number of jobs}.\hfill \\ \begin{array}{c}\mathbf{\text{Step 4. Translate}}\phantom{\rule{0.2em}{0ex}}\text{Write a sentence that gives}\hfill \\ \text{the information to find it.}\hfill \end{array}\hfill & & & \begin{array}{c}\text{\$60 times the number of jobs minus \$1,100 is at least \$4,000.}\hfill \end{array}\hfill \\ \text{Translate into an inequality.}\hfill & & & \hfill 60j-1100 \ge 4,000\hfill \\ \mathbf{\text{Step 5. Solve}}\phantom{\rule{0.2em}{0ex}}\text{the inequality.}\hfill & & & \hfill \phantom{\rule{4.1em}{0ex}}\begin{array}{ccc}\hfill 60j& \ge \hfill & 5,100\hfill \\ \hfill j& \ge \hfill & 85\phantom{\rule{0.2em}{0ex}}\text{jobs}\hfill \end{array}\hfill \\ \begin{array}{c}\mathbf{\text{Step 6. Check}}\phantom{\rule{0.2em}{0ex}}\text{the answer in the problem and}\hfill \\ \text{make sure it makes sense.}\hfill \end{array}\hfill & & & \\ \begin{array}{}\\ \phantom{\rule{2.5em}{0ex}}\text{If Elliot did 90 jobs, his profit would be}\hfill \\ \phantom{\rule{2.5em}{0ex}}60\left(90\right)-1,100,\phantom{\rule{0.2em}{0ex}}\text{or \$4,300. This is}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{more than \$4,000.}\hfill \end{array}\hfill & & & \\ \mathbf{\text{Step 7. Write}}\phantom{\rule{0.2em}{0ex}}\text{a sentence that answers the question.}\hfill & & & \text{Elliot must work at least 85 jobs.}\hfill \end{array}\)

Brenda’s best friend is having a destination wedding and the event will last 3 days. 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?

Solution

\(\begin{array}{c}\mathbf{\text{Step 1. Read}}\phantom{\rule{0.2em}{0ex}}\text{the problem.}\hfill \\ \mathbf{\text{Step 2. Identify}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill & & & \text{the number of hours Brenda must babysit}\hfill \\ \mathbf{\text{Step 3. Name}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill & & & \\ \text{Choose a variable to represent that quantity.}\hfill & & & \text{Let}\phantom{\rule{0.2em}{0ex}}h=\text{the number of hours}.\hfill \\ \begin{array}{c}\mathbf{\text{Step 4. Translate}}\phantom{\rule{0.2em}{0ex}}\text{Write a sentence that}\hfill \\ \text{gives the information to find it.}\hfill \\ \end{array}\hfill & & & \begin{array}{}\\ \\ \text{The expenses must be less than or equal to}\hfill \\ \text{the income. The cost of airfare plus the}\hfill \\ \text{cost of food and entertainment and the}\hfill \\ \text{hotel bill must be less than or equal to the savings}\hfill \\ \text{plus the amount earned babysitting.}\hfill \end{array}\hfill \\ \text{Translate into an inequality.}\hfill & & & \text{\$}350+\text{\$}375+\text{\$}60\left(3\right)\le \text{\$}500+\text{\$}15h\hfill \\ \begin{array}{}\\ \mathbf{\text{Step 5. Solve}}\phantom{\rule{0.2em}{0ex}}\text{the inequality.}\hfill \\ \\ \end{array}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}\begin{array}{}\\ \hfill 905& \le \hfill & 500+15h\hfill \\ \hfill 405& \le \hfill & 15h\hfill \\ \hfill 27& \le \hfill & h\hfill \\ \hfill h& \ge \hfill & 27\hfill \end{array}\hfill \\ \begin{array}{}\\ \mathbf{\text{Step 6. Check}}\phantom{\rule{0.2em}{0ex}}\text{the answer in the problem}\hfill \\ \text{and make sure it makes sense.}\hfill \end{array}\hfill & & & \\ \begin{array}{}\\ \phantom{\rule{2.5em}{0ex}}\text{We substitute 27 into the inequality.}\hfill \\ \phantom{\rule{2.5em}{0ex}}905\le 500+15h\hfill \\ \phantom{\rule{2.5em}{0ex}}905\le 500+15\left(27\right)\hfill \\ \phantom{\rule{2.5em}{0ex}}905\le 905\hfill \end{array}\hfill & & & \\ \mathbf{\text{Step 7. Write}}\phantom{\rule{0.2em}{0ex}}\text{a sentence that answers the question.}\hfill & & & \text{Brenda must babysit at least 27 hours.}\hfill \end{array}\)