General Inequalities and Their Applications

1. A. \(7.50+14 P < 60\), \(24\) Slices

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:

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

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 \($60\).

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

\( 7.50+14 P < 60\)

2) How many pizzas can Jacque afford?

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

\(7.50+14 P < 60\) Subtract \(7.50\)

\(14 P < 52.50\) Divide by \(14\)

\(P < 3.75\)

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.

Let's check our solution

Answers

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

2) Jacque can afford at most \(24\) slices.

2. D. \(24+12 R \geq 100\), \($56\) dollars

Strategy

Sofia needs the sushi she's already ordered plus the additional sushi to be at least \(100\) pieces. We can represent this with an inequality whose structure looks something like this:

\(\text { (sushi already ordered })+\text { (additional sushi) }[\leq \text { or } \geq] 100\)

Then, we can solve the inequality for \(R\) to find how many additional rolls Sofia needs to order.

1) Which inequality?

• Sofia has already ordered and paid for \(24\) pieces.
• Each roll has \(12\) pieces, and \(R\) represents the number of additional rolls, so the number of additional pieces from these rolls is \(12R\).
• The number of pieces she's already ordered plus the additional pieces needs to be greater than or equal to \(100\) pieces.

\(\text { (sushi already ordered) }+\text { (additional sushi) }[\leq \text { or } \geq] 100\)

\(24+12 R \geq 100\)

2) How many additional rolls does Sofia need?

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

\(24+12 R \geq 100\) Subtract \(24\)

\(12 R \geq 76\) Divide by \(12\)

\(R \geq 6 .3\)

Since she can't order partial rolls, Sofia needs to reserve \(7\) additional rolls. And each roll costs \($8\), so ordering \(7\) additional rolls costs \(7 \cdot \$ 8=\$ 56\).

Let's check our solution

Answers

1) The inequality that describes this scenario is \(24+12 R \geq 100\).

2) Sofia needs to spend \($56\) on additional sushi.

3. B. \(34+23 F \geq 175\), \(7\) bags

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:

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

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.

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

\(34+23 F \geq 175\)

2) How many bags does Sergei need?

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

\(34+23 F \geq 175\) Subtract \(34\)

\(23 F \geq 141\) Divide by \(23\)

\(F \geq 6.13 \)

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

Let's check our solution

Answers

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

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

4. \(5+2.75 \cdot S \leq 21\), \(5\) stops

Strategy

The money Julia spends on her ticket must be less than or equal to the \($21\) she has. We can represent this with an inequality whose structure looks something like this:

\(\text { (initial fee) }+\text { (total fees for stops) }[\leq \text { or } \geq] 21\)

Then, we can solve the inequality for \(S\) to find how many stops Julia can afford.

1) Which inequality?

• The initial fee is \($5\).
• Each stop costs \($2.75\) and \(S\) represents the number of stops Julia buys, so she's spending \(2.75 \cdot S\) on stops.
• The combined amount of money she spends on her ticket must be less than or equal \($21\).

\(\text { (initial fee) }+\text { (total fees for stops) }[\leq \text { or } \geq] 21\)

\(5+2.75 \cdot S \leq 21\)

2) How many stops can Julia afford?

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

\(5+2.75 \cdot S \leq 21\) Subtract \(5\).

\(2.75 \cdot S \leq 16\) Divide by \(2.75\)

\(S \leq 5 .81 \)

Since she can't buy partial stops, Julia can afford at most \(5\) stops.

Let's check our solution

Answers

1) The inequality that describes this scenario is \(5+2.75 \cdot S \leq 21\)

2) Julia can afford at most \(5\) stops.