Solving Multi-Step Inequalities

Hi, and welcome to this review of inequalities! Today, we’ll be looking at what inequalities are and how to solve inequality problems. Let’s get started!

Before we dive into inequalities, let’s remind ourselves what an equality is. We know that an “equality,” or an equation, presents values that are equal to each other. An equal sign is the notation that indicates an equation, and it symbolizes the “balancing” of the expressions on either side. In order to solve a linear equation, we use addition, subtraction, multiplication, and division in a specific order to “isolate” the variable on one side of the equation, and a constant on the other side. You have “solved” a linear equation when that variable value is known. It is the one value that, when substituted back into the original equation, will result in a “true” statement.

How to Solve Inequalities

For a quick review, let’s look at the following example:

Solve the equation for \(x\).

\(3x-2=x+8\)

First, we want to get our variables on one side and our constants on another. So I’m going to add 2 to both sides. That gives us:

\(3x=x+10\)

Then, I’m going to subtract \(x\) from this side, and then this side as well (because we want to do the same thing on both sides). That gives us:

\(2x=10\)

And then to get \(x\) by itself divide by 2 on both sides, which gives us:

\(x=5\)

The equation states that the expression on the left side of the equal sign, \(3x-2\), is equivalent to the expression on the right side, \(x+8\). Our goal is to find the one value of \(x\) that makes this statement true.

So now we’re going to check our work; what we’re going to do is we’re going to substitute in the 5 for anywhere we see an \(x\).

\(3(5)-2=15-2=13\)

Now let’s check it on the other side.

\(5+8=13\)

And these two values equal each other.

\(13=13\)

A review of these steps is meaningful, because they remain pretty much the same when solving inequalities. There is a case for some problems that require a small adjustment to the inequality, but we will focus on that later.

It is important to understand that while solving a linear equation results in one solution, solving an inequality results in a set of many solutions. The solving procedure determines the critical value or “boundary” that defines the solution set.

The notation of inequalities determines whether or not the critical value that results from the solving process is included or not included in the solution set. Specifically, the symbols < and > define a solution set that does not include the critical value.

This means that the inequality \(x\)<\(10\) is all the set of numbers less than, but not equal to, 10. Likewise, \(x\)>\(25\) is the set of all numbers greater than, but not equal to, 25.

We can modify this notation slightly to include the critical value that results from the solving process. Note the line under the symbols: \(\leq\) and \(\geq\). Using the previous examples, \(x\leq 10\) is the set of all numbers less than or equal to 10, and \(x\geq 25\) are those numbers greater than or equal to 25.

When inequalities are graphed on a number line, an open circle is used at the critical value to indicate that is NOT included in the solution set, while a closed circle indicates that it is included.

In this example, the closed circle means 4 is included in the solution set:

In this example, the open circle means -2 is not included in the solution set:

Let’s sort out this notation with a few examples of solving inequalities.

\(3x+2 \leq 17\)

The approach here will be the same as solving an equation, but the inequality symbol will be interpreted differently.

The first step to isolate the variable term is to subtract 2 from both sides. This gives us:

\(3x \leq 15\)

The second step is to determine the critical value of \(x\) by dividing both sides by 3:

\(x \leq 5\)

The result, \(x \leq 5\), states that all values of x that are less than or equal to 5 will satisfy the original inequality.

Let’s try another one:

\(24x-1>99-x\)

Remember, the inequality symbol is different in this example, which means that the critical value you will find will not be included in the solution set.

The approach for this multi-step problem is to gather the variable terms to one side, and the constant terms to the other. So we’re going to add 1 to both sides and we’re also going to add x to both sides. This gives us:

\(25x > 100\)

Then, we’ll divide both sides by 25.

\(x> 4\)

This gives us our critical value of 4 because \(x> 4\). This final inequality, \(x> 4\), tells us that all values of x that are greater than but not equal to 4 are in this solution set.

As mentioned, there is a time when you must make an adjustment to the inequality before the solution set is determined. When dividing or multiplying by a negative value in the solving process, it is necessary to reverse the direction of the inequality symbol! This is necessary because sign changes occur with these operations.

The following example illustrates this concept:

\( -5x \geq 20\)

There is only one step necessary in this inequality to determine the critical value. Specifically, we need to divide both sides by -5. Because dividing a signed integer results in a sign change, it is also necessary to reverse the direction of the inequality symbol from \(\geq\) to \(\leq \). So divided by -5 on both sides. Remember to flip your inequality sign and this results in:

\(x\leq -4\)

The solution set is all values of x that are less than or equal to -4.

Let’s see if you can solve this last example on your own. Be sure to:

1. Notice the inequality symbol to determine whether the solution set will include or NOT include the critical value, and
2. Make sure you change the direction of the inequality symbol if you have to multiply or divide by a negative value in the solving process.

\(-12x+2\) < \(-14x -8\)

Pause the video and see what you can come up with.

Okay, let’s solve it together.

Gather the variable terms to one side, and the constant terms to the other. We’re going to subtract 2 from both sides and add \(14x\) to both sides. This leaves us with \(-12x+14x=2x\) is less than \(-8-2=-10\).

\(2x\) < \(-10\)

And then we’re going to divide by 2 on both sides, which gives us \(x\) < \(-5\). Since we divided by +2 and not -2, we didn’t have to flip our inequality sign. So our final solution set is the solution set of all values of \(x\) that are less than but NOT equal to -5. I hope this review of inequalities was helpful! Thanks for watching, and happy studying!

Inequality Problems