Practice Solving Inequality Word Problems

Sergei runs a bakery. He needs at least ‍175 kilograms of flour in total to complete the holiday orders he's received. He only has 34 kilograms of flour, so he needs to buy more.

The flour he likes comes in bags that each contain 23 kilograms of flour. He wants to buy the smallest number of bags as possible and get the amount of flour he needs.

Let \(F\) represent the number of bags of flour that Sergei buys.

1. Which inequality describes this scenario?
   • Choose 1 answer:
     
   1. \(34+23F \leq 175\)
   2. \(34+23F \geq 175\)
   3. \(23+34F \leq 175\)

   4. \(23+34F \geq 175\)

2. What is the smallest number of bags that Sergei can buy to get the amount of flour he needs?

Source: Khan Academy, https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:solve-equations-inequalities/x2f8bb11595b61c86:multistep-inequalities/e/inequalities-solve-problems
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Strategy

The flour Sergei already has plus the flour he buys must be greater than or equal to 175 kilograms. We can represent this with an inequality whose structure looks something like this:

\(\left(  \text{amount he has} \right) + \left(  \text{amount he buys} \right) [\leq \text{or} \geq] \,175\)

Then, we can solve the inequality for \(F\) to find how many bags of flour Sergei needs to buy.

1) Which inequality?

• Sergei already has 34 kilograms of flour.

• Each bag of flour contains 23 kilograms, and ‍\(F\) represents the number of bags he buys, so the amount of flour he buys is \({23 \cdot F}\).

• The amount of flour he has combined with the amount of flour he buys must be greater than or equal to 175 kilograms.

\(\begin{aligned}
\left(  {\text{amount he has}} \right) &+ \left(  {\text{amount he buys}} \right) [\leq \text{or} \geq] \,175
\\\\
{34}&+{23F} {\geq} 175
\end{aligned}\)

2) How many bags does Sergei need?

Let's solve our inequality for \(F\):

\(\begin{aligned}
34+23F &\geq 175 &&\text{Subtract }34
\\\\
23F &\geq 141 &&\text{Divide by }23
\\\\
F &\geq 6.13 \dots
\end{aligned}\)

Since he can't buy a partial bag of flour, Sergei needs to buy 7 bags.

Answers

1. The inequality that describes this scenario is \(34+23F \geq 175\)

2. Sergei needs to buy 7 bags to get the amount of flour he needs.

Nancy needs at least ‍1000 gigabytes of storage to take pictures and videos on her upcoming vacation. She checks and finds that she has 105GB available on her phone. She plans on buying additional memory cards to get the rest of the storage she needs.

The cheapest memory cards she can find each hold 256GB and cost $10. She wants to spend as little money as possible and still get the storage she needs.

Let \(C\) represent the number of memory cards that Nancy buys.

1. Which inequality describes this scenario?
   • Choose 1 answer:
     
   1. \(105+10C \leq 1000\)
   2. \(105+10C \geq 1000\)
   3. \(105+256C \leq 1000\)

   4. \(105+256C \geq 1000\)

2. What is the least amount of money Nancy can spend to get the storage she needs?

Strategy

The storage Nancy has on her phone plus the storage she buys on memory cards needs to be at least ‍1000GB. We can represent this with an inequality whose structure looks something like this:

\(\left(  \text{amount on phone} \right) + \left(  \text{amount from cards} \right) [\leq \text{or} \geq] \,1000\)

Then, we can solve the inequality for \(C\) to find how many memory cards Nancy needs to buy.

1) Which inequality?

• Nancy already has ‍105GB available on her phone.

• Each card has ‍256GB of storage, and ‍\(C\) represents the number of cards she buys, so the amount of memory she buys on cards is \(256C\)

• The amount of memory she has available on her phone combined with the amount of memory she buys from cards must be greater than or equal to 1000GB.

\(\begin{aligned}
\left(  {\text{amount on phone}} \right) &+ \left(  {\text{amount from cards}} \right) [\leq \text{or} \geq] \,1000
\\\\
{105}&+{256C} {\geq} 1000
\end{aligned}\)

2) How many cards does Nancy need?

Let's solve our inequality for \(C\):

\(\begin{aligned}
105+256C &\geq 1000 &&\text{Subtract }105
\\\\
256C &\geq 895 &&\text{Divide by }256
\\\\
C &\geq 3.496 \dots
\end{aligned}\)

Since she can't buy a partial memory card, Nancy needs to buy ‍4 cards. And each card costs ‍$10, so buying ‍4 cards costs ‍\(4 \cdot \$10=\$40\).

Answers

1. The inequality that describes this scenario is \(105+256C \geq 1000\)

2. Nancy needs to spend ‍$40 on memory cards.

Jacque needs to buy some pizzas for a party at her office. She's ordering from a restaurant that charges a ‍$7.50 delivery fee and ‍$14 per pizza. She wants to buy as many pizzas as she can, and she also needs to keep the delivery fee plus the cost of the pizzas under ‍$60.

Each pizza is cut into ‍8 slices, and she wonders how many total slices she can afford.

Let \(P\) represent the number of pizzas that Jacque buys.

1. Which inequality describes this scenario?
   • Choose 1 answer:
     
   1. \(7.50+14P < 60\)
   2. \(7.50+14P > 60\)
   3. \(14+7.50P < 60\)

   4. \(14+7.50P > 60\)

2. What is the largest number of slices that Jacque can afford?

Strategy

Jacque wants the delivery fee plus the cost of the pizzas to be under $60. We can represent this with an inequality whose structure looks something like this:

\(\left(  \text{delivery fee} \right) + \left(  \text{cost of pizzas} \right) [ < \text{or} > ] \,60\)

Then, we can solve the inequality for \(P\) to find how many pizzas Jacque can afford.

1) Which inequality?

• The delivery fee is ‍$7.50.

• Each pizza costs ‍$14, and ‍\(P\) represents the number of pizzas Jacque buys, so the cost of pizzas is ‍\({14 \cdot P}\).

• Jacque wants the delivery fee plus the cost of the pizzas to be under ‍$60, so the total must be less than $60.
  

\(\begin{aligned}
\left(  {\text{delivery fee}} \right) &+ \left(  {\text{cost of pizzas}} \right) [ < \text{or} > ] \,60
\\\\
{7.50}&+{14P} { < } 60
\end{aligned}\)

2) How many pizzas can Jacque afford?

Let's solve our inequality for \(P\):

\(\begin{aligned}
7.50+14P &< 60 &&\text{Subtract }7.50
\\\\
14P &< 52.50 &&\text{Divide by }14
\\\\
P &< 3.75
\end{aligned}\)

Since she can't buy partial pizzas, Jacque can afford at most ‍3 pizzas. And each pizza has ‍8 slices, so buying ‍3 pizzas gets her ‍\(3 \cdot 8=24\) slices.

Answers

1. The inequality that describes this scenario is \(7.50+14P < 60\)

2. Jacque can afford at most 24 slices.
   

Alonso went to the market with ‍$55 to buy eggs and sugar. He knows he needs a package of 12‍ eggs that costs ‍$2.75. After getting the eggs, he wants to buy as much sugar as he can with his remaining money. The sugar he likes comes in boxes that each cost ‍$11.50.

Let \(S\) represent the number of boxes of sugar Alonso buys.

1. Which inequality describes this scenario?
   • Choose 1 answer:
     
   1. \(2.75+S \leq 55\)
   2. \(2.75+S \geq 55\)
   3. \(2.75+11.50S \leq 55\)

   4. \(2.75+11.50S \geq 55\)

2. After getting the eggs, how many boxes of sugar can Alonso afford?

Strategy

The money Alonso spends on eggs plus the money he spends on sugar must be less than or equal to ‍$55. We can represent this with an inequality whose structure looks something like this:

\(\left(  \text{money on eggs} \right) + \left(  \text{money on sugar} \right) [\leq \text{or} \geq] \,55\)

Then, we can solve the inequality for \(S\) to find how many boxes of sugar Alonso can afford.

1) Which inequality?

• The package of eggs costs ‍$2.75, and Alonso needs one package, so he's spending ‍$2.75on eggs.

• Each box of sugar costs ‍$11.50, and ‍\(S\) represents the number of boxes of sugar he buys, so he's spending ‍\({11.50 \cdot S}\) on sugar.

• The combined amount of money he spends on eggs and sugar must be less than or equal to $55.

\(\begin{aligned}
\left(  {\text{money on eggs}} \right) &+ \left(  {\text{money on sugar}} \right) [\leq \text{or} \geq] \,55
\\\\
{2.75}&+{11.50S} {\leq} 55
\end{aligned}\)

2) How many bags does Sergei need?

Let's solve our inequality for \(S\):

\(\begin{aligned}
2.75+11.50S &\leq 55 &&\text{Subtract }2.75
\\\\
11.50S &\leq 52.25 &&\text{Divide by }11.50
\\\\
S &\leq 4.54 \dots
\end{aligned}\)

Since he can't buy partial boxes of sugar, Alonso can afford at most ‍4 boxes of sugar.

Answers

1. The inequality that describes this scenario is \(2.75+11.50S \leq 55\)

2. After getting the eggs, Alonso can afford ‍4 boxes of sugar.