Inequalities, like equations, can be translated and used to solve problems. We use an inequality for values that aren't the same, or that can only be the same up to a certain amount.
When a problem says "at least" or "no less than," this means the number given is the very smallest it'll go; it can't get any smaller. The remaining value must be something bigger. For example, if a bag of candy has at least 28 pieces, we know it has 28 or more pieces. So the number of pieces (x) is greater than or equal to 28.
x ≥ 28
The phrases "at most" and "no more than" both mean the number given is the biggest the value will ever get. If a box of drinks has no more than 15 drinks, it has 15, or 14, or 13…drinks. Notice these values are all less than or equal to 15.
x ≤ 15
Sometimes we're lucky and the problem just uses the phrase "less than" (<) or "greater than" (>). But humans are not robots—we like to get creative about our phrasing—so don't count on that.
Example 1: One-Step Inequalities
A bag of candy is split between us and our little brother. The bag says it has at most 28 pieces of candy in it. There are 15 candies in our bag, and x candies in our brother's bag, and we have to make sure he doesn't have more than us (of course). We can write an inequality expressing the number of candies in the bags.
We know that our bag has 15 candies.
We know that his bag has x candies.
We know that up to 28 candies were split between both bags, no more.
Since we know that x + 15 can't be more than 28, it must be less than or equal to 28.
x + 15 ≤ 28
Now that we have an inequality expressing our candy, we solve for x. To do that, we subtract 15 from both sides.
x + 15 ≤ 28
x + 15 – 15 ≤ 28 – 15
x ≤ 13
Sweet, literally. Our brother has 13 or fewer pieces of candy, so no way does he have more candy than us. Until, in a moment of weakness (er, kindness) we share a few of our pieces with him. Now it's even sweeter.
Example 2: Two-Step Inequalities
A candy store owner saw us share our candy with our brother and was so impressed he gave us a $30.00 gift card. Tax free even! We decide to buy a giant candy bar for $13.00 and then some lollipops with the remaining money. If each lollipop is $0.90, we can write and solve an inequality expressing how many lollipops we can buy.
We know that the amount we spend needs to be less than or equal to $30.00.
We know we want to buy x lollipops.
We know that each lollipop is $0.90, so the total cost of the lollipops is $0.90x.
We know the candy bar costs $13.00.
So the total cost of the candy is $13 + $0.90x, and this needs to be less than or equal to $30.00.
13 + 0.90x ≤ 30
Now we solve for x, which is the number of lollipops we can buy. We want to get x by itself, so start by subtracting 13 from both sides.
13 + 0.90x ≤ 30
13 + 0.90x – 13 ≤ 30 – 13
0.9x ≤ 17
Divide both sides by 0.9 to finish up. We recommend nabbing a calculator for this part.
We need to buy less than 18.89 lollipops, so we can buy 18 whole lollipops and have some change to spare.
Example 1
Christy is playing an online game mining precious jewels, and she comes across a mother lode of emeralds. The most she's ever mined is 4 blocks of emeralds. Write an inequality expressing how many blocks she needs to mine to beat her record. |
Example 2
Farmer Ted is putting up a chain link fence around a rectangular garden plot. He has 75 feet of fence and wants the garden to be 15 feet wide. Write an inequality expressing the possible length of the garden plot. |
Example 3
Alex sells books online. She makes a flat profit of $2.00 per book, but she needs to pay $4.00 per day to Paypal for using the app on her website. How many books does she need to sell to make at least $120.00 per day? |