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 𝑎𝑥+𝑏<𝑐. Consider the inequality 4𝑥+824.

Our first step in solving this is to subtract eight from both sides: 4𝑥16.

We then divide both sides by four to get 𝑥4.

We can represent this solution on a number line:

Or as an interval: [4,).

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 7𝑥+22𝑥+7.

This question is asking for what values of 𝑥 the line 𝑦=7𝑥+2 is less than the line 𝑦=2𝑥+7. We can see the solution to this if we draw the two lines on a graph:

The green line is 𝑦=7𝑥+2 and the blue line is 𝑦=2𝑥+7. We can see that the green line is less than or equal to the blue line when 𝑥1, which is the solution to the inequality.

Let us look now at how we can solve this algebraically. Consider again the inequality 7𝑥+22𝑥+7.

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 2𝑥 here, from each side: 5𝑥+27.

Then, we subtract two from each side: 5𝑥5.

Finally, we divide both sides by five to get 𝑥1.

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.

Example 1: Solving Linear Inequalities with a Variable on Each Side

Solve the following inequality: 9𝑦>77+2𝑦.

Answer

First, we need to subtract 2𝑦 from each side to get 7𝑦>77.

Then, we need to divide both sides by seven to get 𝑦>11.

We could also write our answer as an interval: (11,).

Now, let us look at an example containing negative coefficients.

Example 2: Solving Linear Inequalities with a Variable on Each Side

Solve the following inequality: 9𝑥+14208𝑥.

Answer

Firstly, to avoid having to divide by a negative number, it is easiest to add 9𝑥 to both sides of the inequality: 1420+𝑥.

Then, we need to add 20 to both sides to get 34𝑥.

So, 𝑥 is less than or equal to 34. We can equally write this as 𝑥34, and we could also write our answer as an interval: (,34].

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