Solving Multi-Step Inequalities with Decimals

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  • 0:01 A Multi-Step Inequality
  • 1:42 Making a Profit
  • 4:00 Negative Signs
  • 4:35 Losing Money
  • 6:57 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this video lesson, you will learn the steps that are involved in solving inequalities for an unknown variable. We'll see that the order of operations stays the same, but there's a very significant difference when it comes to negative numbers!

A Multi-Step Inequality

In this video lesson, we will take a look at how to solve inequalities with decimals in them. Let's see how Sam, who owns a skateboard shop, handles these types of problems. Every month, Sam sits down and does math to see how well his shop is doing. The kind of math he does helps him to see if he is charging the right price for his skateboards. For example, this problem helps him to see if he will make money off of selling his skateboards or not.

10(x - 25.25) ≥ 100

With this problem, Sam can figure out at what price to sell his skateboards so that he can make money instead of losing money. The way this particular problem is set up helps Sam to see what kind of pricing he needs so that he will make a profit of at least $100 when he sells 10 skateboards.

The 25.25 stands for how much it costs Sam to make one skateboard. The x, our variable, stands for the cost of one skateboard. This type of problem that Sam uses is called an inequality. It is an equation that uses an inequality sign instead of an equal sign.

See how this inequality will give us a choice of pricing for the skateboards? Once it is solved, Sam will know in what range to price his skateboards. Let's see how Sam goes about and solves this type of inequality. Also notice that we have decimals, numbers with a decimal point. This doesn't deter Sam from solving the problem at all. Let's watch.

Making a Profit

Sam sees that there is a set of parentheses in his problem, so he tackles that first. Sam is following the order of operations that is universal for any type of problem in math. He goes ahead and performs the multiplication. He multiplies the 10 with the x first and then the -25.25. He gets:

10x - 252.5 ≥ 100

Now that the parentheses are taken care of, Sam looks for any addition or subtraction that is taking place with his variable. He sees there is subtraction going on. So, he goes ahead and performs the opposite operation, addition, to move what is being subtracted away from the x and to the other side.

In math, this is known as performing the inverse operation, since addition is the inverse operation of subtraction. If Sam had seen addition, then he would have performed subtraction from both sides of the inequality. Either way, to keep the equation the same, Sam makes sure he performs the same operation to both sides of his inequality. He gets:

10x - 252.5 + 252.5 ≥ 100 + 252.5

which turns into

10x ≥ 352.5

Sam's next step is to look for any multiplication or division going on with the variable. Sam sees that the variable is being multiplied by 10. Sam does what he did before and performs the opposite operation to separate this number from the variable and move it to the other side. Again, he performs the same operation to both sides of the inequality. He gets:

10x/10 ≥ 352.5/10

Evaluating this gives Sam an answer of:

x ≥ 35.25

This tells Sam that as long as he sells his skateboards for more than $35.25 each, then he will make at least $100 profit when he sells 10 of them.

Negative Signs

Before we continue, I have to point out a very interesting thing about working with inequalities. If we ever need to divide or multiply by a negative number, then our inequality sign will flip. For example, a greater than sign will flip into a less than sign and vice versa.

So, if Sam were to multiply by a -10, then he would change his greater than inequality sign to a less than inequality sign. On the other hand, if Sam had a less than sign, then he would change it to a greater than sign.

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