# Less Than Symbol in Math: Problems & Applications

## The Language of Math

Mathematics is a language that uses many symbols. Learning all the symbols can be tricky, but it's critical for understanding what is being communicated, just like learning words and the rules for grammar is key to speaking and understanding a foreign language, like Spanish.

## Greater Than or Less Than

When two things are the same in every way, they are said to be **equal**. When two values are not the same, there is an **inequality**. In math, problems most often center around equations of numbers that are either equal or unequal. If the equations or numbers are not equal, it stands to reason that one of them is bigger than the other.

The **less-than symbol** (< ) is used to signify that the number on the left is smaller, or less, than the number on the right. The **greater-than symbol** (>) is used to signify that the number on the left is larger, or greater, than the number on the right. The less-than and greater-than symbols are actually the same symbol, the direction of which is switched depending on whether the number on the left is larger or smaller.

You may remember learning to use these symbols with the aid of an alligator when you were younger. The alligator is hungry, and so he opens his mouth towards the bigger number.

## Solving Inequalities

The less-than symbol, as well as the greater-than symbol, can be used for more than just showing which number is larger than another. In fact, many inequalities require you to solve the problems on each side of the less-than symbol in order to determine the relative value of a variable. Here is an example of a math problem containing the less-than symbol:

3*x* + 2 < *x* - 4

Just like you would if there was an equal sign, you want to group like terms together. Subtracting an *x* from either side, we get:

2*x* + 2 < -4

Subtract a 2 from either side to get:

2*x* < -6

Finally, dividing by 2, we have:

*x* < -3

You can see that solving an inequality is very similar to solving an equation. One important difference is our solution is not a single value. In this example, our solution includes all values less than -3.

## Applications

Inequalities are common in everyday life. Many real-world examples of problems involving inequalities have to do with money. Sometimes you need to determine if you have enough money to purchase a specific item, or you may need to determine how much of something you can buy with the money you have. Here is an example in which understanding the less-than principle comes in handy:

Max wants to buy some t-shirts online. The t-shirts are $10 each and shipping will be $15. He only has $100 available to spend. How many t-shirts can he buy and stay within his $100 limit?

To solve this inequality, you first need to write an equation. Let *x* equal the number of t-shirts and the inequality should look like this:

10*x* + 15 < 100

This inequality states that $10 times the number of shirts plus $15 for shipping has to be less than the $100 Max has available to spend.

If you solve the inequality, you'll come to the solution *x* < 8.5. This means that the number of t-shirts Max can buy will be less than 8.5; he can purchase 8 t-shirts under his budget (since he cannot buy a fraction of a t-shirt). This is a simple example, but more complicated less-than/greater-than problems are tackled every day by finance professionals, architects, builders and people just living their daily lives.

## Lesson Summary

The **less-than symbol** (<) represents an **inequality** in which the number or problem on the left is smaller than that on the right; however, it can be reversed (>) to show that the item on the left is greater than that on the right. These inequalities have everyday applications in a wide variety of areas, especially when dealing with money.

## Learning Outcomes

View this video lesson, then assess your capacity to:

- Identify and explain the use of the inequality symbol for less than
- Distinguish between the less than and greater than symbols
- Solve an inequality that includes the less than symbol
- Recognize the real-life applications of inequalities
- Turn an inequality word problem into a math equation

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## Additional Application Problems

Applications in mathematics are often shown using word problems. Many students struggle with rewriting the words into math. In the following examples, students will practice translating real-world situations into inequalities and then will solve the inequalities.

## Examples

1) You want to purchase some DVDs from a website. Each DVD costs $12.00, and shipping costs $6.99. You have $50.00 that you can spend. Write an inequality representing the situation and solve for the number of DVDs you can purchase.

2) Your favorite cereal is on sale at the grocery store for $2.25 per box. Sales tax is 6.5%, and you have $15.00. Write an inequality representing the situation and solve for the number of boxes you can buy.

3) A car rental costs $50.00 plus an additional $1.25 per mile driven. If you want to spend less than $95.00 for a rental, write an inequality and solve it to find the number of miles you can drive.

## Solutions

1) If *x* stands for the number of DVDs you can buy, we can write the inequality 12*x* + 6.99 < 50, meaning that 12 dollars multiplied by the number of DVDs plus the shipping cost of $6.99 must be less than $50.00. To solve the inequality, subtract 6.99 from both sides and divide by 12.

So you can buy at most 3 DVDs.

2) If *x* stands for the number of boxes of cereal, we can write 2.25*x* + 0.065 * 2.25*x* < 15. The sales tax of 6.5% is calculated by multiplying the cost of *x* boxes of cereal by 0.065. Then, simplifying and solving the inequality:

So you can buy at most 6 boxes of cereal.

3) If *x* stands for the number of miles driven, then the car rental can be represented by 50 + 1.25*x* < 95. Solving this inequality, we have:

So you must drive less than 36 miles.

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