SAT Linear Inequalities Question 110 | Free SAT Prep | Aniko

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Question 110·Medium·Linear Inequalities in One or Two Variables

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At a school fundraiser, the club earns $12 for each bracelet and $20 for each hoodie sold. The club needs to raise at least $600. If $b$ is the number of bracelets and $h$ is the number of hoodies sold, which inequality represents all combinations of $b$ and $h$ that meet the fundraising goal?

For SAT word problems with inequalities, first translate each part of the story into algebra: write an expression for the total quantity (here, total money: $12b + 20h$ ) by multiplying the per-item amount by the number of items and adding. Then translate key phrases precisely: “at least” means $\ge$ , “at most” means $\le$ , “more than” means $>$ , and “less than” means $<$ . Finally, place the inequality symbol between your total expression and the target number and quickly check that larger totals are allowed or disallowed in a way that fits the wording.

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Represent the total money in terms of b and h

First, write an expression for how much money comes from bracelets and another for how much comes from hoodies, then add them.

Focus on the phrase “at least $600”

Does “at least $600” mean the total money should be less than 600, more than 600, or 600 or more? Which inequality symbol matches that idea?

Form the inequality

Once you have the expression for total money and you know which direction the inequality should go, place the inequality sign between the total expression and 600.

Desmos Guide

Choose test combinations of bracelets and hoodies

Pick one combination that clearly meets the goal and one that clearly does not. For example:

A meeting combo: 10 bracelets and 25 hoodies gives $12(10) + 20(25) = 120 + 500 = 620$ , which is at least $600.
A non-meeting combo: 5 bracelets and 10 hoodies gives $12(5) + 20(10) = 60 + 200 = 260$ , which is less than $600.

In Desmos, plot these as points $(10,25)$ and $(5,10)$ .

Graph each answer choice as an inequality

Treat $x$ as $b$ (bracelets) and $y$ as $h$ (hoodies). For each answer choice, type the inequality exactly as written but replace $b$ with $x$ and $h$ with $y$ (for example, for choice A, type 12x + 20y <= 600). Desmos will shade the solution region for that inequality.

Check which inequality matches the situation

For each graphed inequality, see whether the point that meets the goal $(10,25)$ lies in the shaded region and the point that does not meet the goal $(5,10)$ lies outside it. The correct answer choice is the one whose shaded region includes the successful combination and excludes the unsuccessful one.

Step-by-step Explanation

Write expressions for the money from each item

Each bracelet earns $12, and there are $b$ bracelets, so the money from bracelets is $12b$ .

Each hoodie earns $20, and there are $h$ hoodies, so the money from hoodies is $20h$ .

Combine to get the total amount of money raised

To find the total money raised, add the money from bracelets and hoodies:

\text{total money} = 12b + 20h

This expression represents the total dollars the club earns from selling $b$ bracelets and $h$ hoodies.

Translate “at least $600” into an inequality and match the choice

The phrase “at least $600” means the total money is 600 or more, so the total must be greater than or equal to 600. In symbols, that is written with $\ge 600$ .

So we require

12b + 20h \ge 600

This matches answer choice D) $12b + 20h \ge 600$ .

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation