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Lesson Explainer: Linear Inequalities

In this explainer, we will learn how to solve linear inequalities and two linear inequalities combined using “and” or “or” in one variable.

We can solve linear inequalities using very similar techniques to those of solving linear equations. Recall that we can solve a linear equation by isolating the variable (typically 𝑥) on one side of the equation. For instance, consider the equation 2𝑥+7=3.

By subtracting 7 from both sides, we get 2𝑥=4.

Then, by dividing both sides by 2, we have 𝑥=2.

Thus, we have shown that the solution is 𝑥=2.

Let’s suppose instead that we had an inequality sign (e.g., “”) in place of the equal sign: 2𝑥+73.

Once again, we can subtract 7 from both sides: 2𝑥4.

Then, we can divide both sides by 2: 𝑥2.

We can see that the steps for solving the inequality are exactly the same. We just rearrange the inequality so that we can isolate the variable 𝑥. The main difference is that the solution to a linear inequality is itself in the form of an inequality rather than a single value. Thus, the solution is actually a set of values for which the inequality is valid.

The difference can be appreciated by considering a number line.

Any value of 𝑥 greater than or equal to 2 satisfies the inequality 2𝑥+73; however, only the value of 𝑥=2 is a solution to the equation 2𝑥+7=3.

Let’s work through a similar example of solving a straightforward linear inequality.

Example 1: Solving an Inequality

Find the set of values of 𝑥 that satisfy the inequality 3𝑥+72.

Answer

To solve an inequality in 𝑥, we want to isolate 𝑥 on one side of the inequality by applying the same operations to both sides of the inequality. We can start by subtracting 7 from both sides of the inequality to obtain 3𝑥9.

We now want to divide through by 3; note that since this value is positive, we do not need to switch the direction of the inequality. We have 3𝑥393𝑥3.

Hence, the set of values of 𝑥 that satisfy the inequality is 𝑥3.

It is very important to consider the direction of the inequality after applying operations to both sides of the inequality transformations.

For example, imagine we want to solve 𝑥>1. We can do this by asking which values are greater than 1 after being multiplied by 1. We can note that (2)×(1)=2, and so 2 satisfies the inequality. In fact, any value below 1 will satisfy the inequality.

We can justify why this happens by considering a geometric interpretation. Multiplying a number by 1 on a number line switches its sign; it is the same as reflecting the number line through 0.

This tells us that multiplying both sides of an inequality by a negative number will switch the direction of the inequality since it involves a reflection. We have the following properties that we can use to solve inequalities.

Property: Properties of Inequalities

  • We can add any value to or subtract any value from both sides of an inequality to obtain an equivalent inequality. This can also be a variable.
  • We can multiply or divide both sides of an inequality by a positive constant to obtain an equivalent inequality.
  • We can multiply or divide both sides of an inequality by a negative constant to obtain an equivalent inequality by reversing the direction of the inequality.

We can use these properties to solve linear inequalities by isolating the variable on one side of the inequality.

In our next example, we will solve an inequality with 𝑥-terms on either side.

Example 2: Solving an Inequality with Multiple 𝑥-Terms

Find the set of values of 𝑥 that satisfy the inequality 4𝑥+5>2𝑥+1.

Answer

To solve an inequality in 𝑥, we want to isolate 𝑥 on one side of the inequality by applying the same operations to both sides of the inequality. There are a few different ways we could do this.

The first method involves adding 4𝑥 to both sides of the inequality to obtain 5>2𝑥+1.

We can then subtract 1 from both sides of the inequality to get 4>2𝑥.

We can then divide the inequality through by 2; we note that this is positive, so the direction of the inequality is maintained. We have 2>𝑥.

We usually write inequalities with the variable on the left, and we can do this by switching the direction of the inequality and swapping the sides to get 𝑥<2.

The second method involves adding 2𝑥 to both sides of the inequality to obtain 2𝑥+5>1.

We then subtract 5 from both sides of the inequality to get 2𝑥>4.

We now need to divide both sides of the inequality by 2 to isolate 𝑥. We recall that since this value is negative, we also need to switch the direction of the inequality. We get 𝑥<2.

Hence, the set of values of 𝑥 that satisfy the inequality is 𝑥<2.

In our next example, we will solve an inequality by expanding the brackets and simplifying.

Example 3: Solving an Inequality by Expansion

Find the set of values of 𝑥 that satisfy the inequality 2(3𝑥2)(2𝑥+4).

Answer

To solve an inequality in 𝑥, we want to isolate 𝑥 on one side of the inequality by applying the same operations to both sides of the inequality. In order to do this, we first need to expand the brackets on both sides of the inequality. We do this by distributing each factor:

We can then solve the inequality by isolating the variable. We add 2𝑥 to both sides of the inequality to obtain 8𝑥44.

We now add 4 to both sides of the inequality to get 8𝑥0.

We now divide both sides of the inequality by 8 and note that this is a positive value, so we do not need to reverse the inequality. We obtain 𝑥0.

Hence, the set of values of 𝑥 that satisfy the inequality is 𝑥0.

In our next example, we will solve an inequality by expanding the brackets and simplifying.

Example 4: Solving an Inequality by Expansion

Find the set of values of 𝑥 that satisfy the inequality 𝑥(𝑥+3)𝑥(6𝑥)+3.

Answer

To solve an inequality in 𝑥, we want to isolate 𝑥 on one side of the inequality by applying the same operations to both sides of the inequality. In order to do this, we first need to expand the brackets on both sides of the inequality:

We now want to combine the like terms by rewriting the inequality. We can add 𝑥 to both sides of the inequality to get 3𝑥6𝑥+3.

We can now add 3𝑥 to both sides of the inequality. This gives us 09𝑥+3.

We now want to isolate 𝑥, so we subtract 3 from both sides of the inequality: 39𝑥, and then we divide both sides of the inequality by 9; we note that this is a positive value, so we do not need to reverse the inequality. We obtain 13𝑥.

We usually write the variable on the left-hand side of the inequality. We can do this by switching the direction of the inequality and swapping the sides to get 𝑥13.

Hence, the set of values of 𝑥 that satisfy the inequality is 𝑥13.

Sometimes, we will want to solve multiple linear inequalities. We can do this by solving each linear inequality and then combining the solution sets at the end. It is also a good idea to sketch the solution set to each inequality on a number line to get a good idea of exactly what is happening.

For example, let’s say we want to find the values of 𝑥 that solve both 𝑥+1>2 and 2𝑥6. We start by solving each inequality by using the properties of the inequality. We subtract 1 from both sides of the first inequality to get 𝑥>1.

We then divide both sides of the second inequality by 2 to obtain 𝑥3.

We want the values that satisfy both inequalities; we can find this by sketching both solution sets on a number line.

We want the values that are in both solution sets, so they must be overlapping on the number line.

We see that this is all the values between 1 and 3, including 3. We can write this as the compound inequality 1<𝑥3, which is really two separate inequalities: 1<𝑥 and 𝑥3.

We can write this in set notation in two different ways. We can use the compound inequality to get {𝑥1<𝑥3}. However, we can also use the individual inequalities and the fact that the intersection of sets means all the values in both sets to write this as {𝑥𝑥>1}{𝑥𝑥3}. It is also worth noting that if there is no overlap between the two inequalities, then there are no values that satisfy both inequalities.

In a similar way, we could also find the solutions to either of two inequalities. Let’s say we want to represent the solutions to either 𝑥>3 or 𝑥0. We can sketch these sets on a number line.

We see that there is no overlap, so we must leave these as separate inequalities. We can also write this in set notation by recalling that the union of two sets gives us all the values in either set. Therefore, the solutions to either inequality are given by the set {𝑥𝑥0}{𝑥𝑥>3}.

In our next example, we will find and represent the solution to either of two given inequalities by using set notation.

Example 5: Finding the Solutions to Either of Two Inequalities

Find the set of values of 𝑥 that satisfy either the inequality 2(𝑥1)3(𝑥5) or the inequality 5(𝑥+4)3(32𝑥). Give your answer in set notation.

Answer

We first note that we are looking for the values of 𝑥 that solve either of the two given inequalities. Therefore, we can start by finding the solutions of each inequality separately and then combining the solution sets afterward.

Let’s start by solving 2(𝑥1)3(𝑥5). We expand the brackets on both sides of the inequality to obtain 2𝑥23𝑥15.

We now want to isolate 𝑥 on one side of the inequality; we can do this by subtracting 2𝑥 from both sides of the inequality to get 2𝑥15.

We then add 15 to both sides of the inequality. This gives us 13𝑥.

We usually write the variable on the left-hand side of the inequality. We can do this by switching the direction of the inequality and swapping the sides to get 𝑥13.

We can follow a similar process to find the solution set of the other inequality. We start by expanding the brackets to obtain 5𝑥+2096𝑥.

We then add 6𝑥 to both sides of the inequality to get 11𝑥+209.

We now subtract 20 from both sides of the inequality. This gives us 11𝑥11.

We now divide both sides of the inequality by 11; we note that this is a positive value, so we do not need to reverse the inequality. We obtain 𝑥1.

We have found that the values of 𝑥 that satisfy one of the two inequalities are the values of 𝑥 that satisfy 𝑥13 or 𝑥1. In order to help us visualize this set, we can represent both of the inequalities on a number line.

We recall that we use a solid dot to show that the endpoint is included and an arrow to show the direction of the inequality. We get the following.

There is no overlap in the two sets, so we have to leave these inequalities separate.

We are told to represent the solution set by using set notation. We can represent each individual inequality in set notation. We have {𝑥𝑥1},{𝑥𝑥13}.

Finally, we want to say that we can take any value of 𝑥 in either of these sets. We recall that we do this by taking their union: {𝑥𝑥1}{𝑥𝑥13}.

In our final example, we will find and represent the solution to two given inequalities by using set notation.

Example 6: Finding the Solution Set to Two Inequalities

Find the set of values of 𝑥 that satisfy the inequalities 3(𝑥1)2(𝑥1) and 5(𝑥+3)>2(53𝑥). Give your answer in set notation.

Answer

We first note that we are looking for the values of 𝑥 that solve both of the two given inequalities. In order to do this, we first need to solve each inequality separately and then find the values inside both solution sets.

Let’s start by solving 3(𝑥1)2(𝑥1). We expand the brackets on both sides of the inequality to obtain 3𝑥32𝑥2.

We now want to isolate 𝑥 on one side of the inequality; we can do this by subtracting 2𝑥 from both sides of the inequality to get 𝑥32.

We then add 3 to both sides of the inequality. This gives us 𝑥1.

We can follow a similar process to find the solution set of the other inequality. We start by expanding the brackets to obtain 5𝑥+15>106𝑥.

We then add 6𝑥 to both sides of the inequality to get 11𝑥+15>10.

We now subtract 15 from both sides of the inequality. This gives us 11𝑥>5.

We now divide both sides of the inequality by 11; we note that this is a positive value, so we do not need to reverse the inequality. We obtain 𝑥>511.

We have found that the values of 𝑥 that satisfy one of the two inequalities are the values of 𝑥 that satisfy 𝑥1 and 𝑥>511. In order to help us visualize this set, we can represent both inequalities on a number line.

We recall that we use a solid dot to show that the endpoint is included, a hollow dot to show that the endpoint is not included, and an arrow to show the direction of the inequality. We get the following.

We want the values that are in both solution sets. In other words, they are overlapping. We can see from the diagram that this is all the values between 511 and 1, and 511 is not included.

We can write this as an inequality: 511<𝑥1. However, we are asked to write this in set notation. Thus, the solution set is 𝑥511<𝑥1.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • Adding or subtracting a constant or algebraic expression from either side of an inequality gives an equivalent inequality.
  • Multiplying or dividing both sides of an inequality by a positive number gives an equivalent inequality.
  • Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality.
  • We can represent solutions to inequalities on a number line by using a solid dot to represent an endpoint being included in the solution set, a hollow dot to represent an endpoint not being included in the solution set, and an arrow to show the direction of the solution set if it is unbounded.
  • We can also represent solutions to multiple inequalities by using set notation. We can use the union of two solution sets to show that a value from either set will satisfy one or more inequalities, and we can use the intersection of two solution sets to show that a value must be in both sets to satisfy one or more inequalities.

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