Lesson Explainer: Two-Step Linear Inequalities

In this explainer, we will learn how to solve a linear inequality in two steps.

Let us start by recapping how to solve linear inequalities in the form . Consider the inequality

Our first step in solving this is to subtract eight from both sides:

We then divide both sides by four to get

We can represent this solution on a number line:

Or as an interval: .

Remember here that we have drawn a shaded circle to show that the 4 is included in the number line and we have used a square bracket in the interval for the same reason.

Now, let us have a look at inequalities that contain a variable on each side. Consider the inequality

This question is asking for what values of the line is less than the line . We can see the solution to this if we draw the two lines on a graph:

The green line is and the blue line is . We can see that the green line is less than or equal to the blue line when , which is the solution to the inequality.

Let us look now at how we can solve this algebraically. Consider again the inequality

We can solve this by taking a very similar approach to that of solving equations with a variable on each side by applying inverse operations. First, we subtract the variable with the smallest coefficient, which is here, from each side:

Then, we subtract two from each side:

Finally, we divide both sides by five to get

One thing to note is that care needs to be taken when dealing with negative coefficients. If you multiply or divide an inequality by a negative number the direction of the inequality switches. To avoid this you can rearrange the inequality to make the coefficients positive.

Let us look now at a formal example.