How to Solve Inequalities Using Estimation

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  • 0:02 Estimations
  • 0:31 Inequality Review
  • 1:42 Estimating Inequalities
  • 5:47 Tips & Warnings
  • 6:46 Lesson Summary
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Lesson Transcript
Instructor: Robert Egan
In this lesson, you'll learn how to solve inequality problems without doing a lot of complicated addition and subtraction. We'll explore the process of estimating the quantities on both sides of the inequality symbol, which can be helpful when solving these problems.


How many marbles are in this jar? You don't know, but can you guess? Is it closer to 10 or to 100? 100, right? It's probably even more than that.

You don't know how many marbles are in this jar, either. But which jar has more, the big one or the small one? Obviously, it's the big one! Congratulations, you just solved an inequality using estimation. In this lesson, you'll learn some more about what it means, and how to do it with numbers, not just marbles.

Inequality Review

An inequality is an equation where you have to tell which quantity is bigger than the other. We write inequalities using inequality symbols: greater than (>), less than (<), and equal to (=). You can remember which way the signs go if you think of them as alligator mouths. They're very greedy alligators, so they always want to eat the bigger number.

When you work with inequality problems, you sometimes get problems like this:

5 + 2 ____ 3 + 1

Your job is to decide which symbol goes in the blank, so you have to figure out which side of the inequality is bigger. Normally, you would do this by adding up both sides:

5 + 2 ____ 3 + 1

7 ____ 4

7 > 4

7 is greater than 4, so the answer here would be the greater-than symbol. That one was pretty easy. But, what if you get something like this:

347 + 584 + 68 ____ 415 + 186 - 52

You could definitely solve this one by doing the math on both sides. But that's some pretty ugly math, and there's an easier way: estimation.

Estimating Inequalities

Estimation is determining the approximate value of something in a way that's good enough for whatever you want to do with it. For example, the U.S. government cites the population of the United States in 2015 as 321,368,864 people and the population of Canada as 35,099,836. But, on any given day, those numbers will be very slightly different because people are always being born and dying, so the populations are always changing.

If a newspaper wanted to compare the population of the U.S. to the population of Canada, it would have a really tough job figuring out exactly how many people live in each country because that's changing every day! But, is it really important to know down to each individual person?

Just to compare the two countries, the paper might report 'approximately 321 million people' in the U.S. and 'approximately 35 million people' in Canada. Those numbers are easier to think about and they're close enough to give you a basically accurate idea of the comparison.

That example illustrates how you could use estimated numbers to get a 'good enough' answer for the comparison without wasting time and energy sweating over the details. Remember that you don't need to know the exact value of each thing in the inequality. Just like the jars of marbles, you only need to know which is bigger.

Sometimes it's easy to figure that out without doing any math. Just, for example, let's say you had an inequality problem like this:

85 + 23 + 42 ____ 5 + 3 + 2

You can tell right away which side will be bigger without doing any addition at all. On the left side, you're adding three big numbers. On the right side, you're adding three small numbers. You don't have to actually add 85 + 23 + 42 to know that it's going to be much bigger than 5 + 3 + 2!

That one really was like the jars of marbles - you could tell just by looking at it! But even if you can't figure it out that easily, you can use estimation to help you solve inequality problems by rounding numbers without actually having to do all the math. Let's take our example from before:

584 + 347 + 68 ____ 415 + 186 - 52

You can estimate the quantities on each side by rounding all the numbers to the hundreds' place. If you just look at the hundreds' place of all these numbers, you can see that you've got 600 + 300 + 100 compared to 400 + 200 - 100. That gives you approximately 1,000 on the left and 500 on the right. This is a very rough estimation of how big the numbers are on each side.

You can see right away that the left-hand side looks bigger, but let's check to make sure we haven't introduced any strange mistakes from the rounding. Since we're rounding to the hundreds' place, the biggest possible difference between the original number and the rounded number will be 50.

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