In this explainer, we will learn how to solve compound linear inequalities by applying inverse operations.
Inequalities are useful for giving us information about how two quantities are related when they are not necessarily equal. Often, a single inequality does not give us enough information regarding the relationship between two quantities; in such cases, we might use compound inequalities. In general, compound inequalities give us two or more conditions that hold either separately or simultaneously.
Compound Inequality
A compound inequality can be presented in two forms. It can be two inequalities connected by an “and,” which are often presented together and can be separated into two separate inequalities, an example being the compound inequality
This is an amalgamation of the inequalities and
Alternatively, compound inequalities can be connected by an “or”; for example,
Here, the variable needs to satisfy either the first or the second inequality.
To solve a compound inequality, we often solve each inequality separately and then combine the solutions. We will begin by looking at the inequalities introduced above.
Example 1: Solving Compound Linear Inequalities
Find all the values of that satisfy . Give your answer in interval form.
Answer
We can solve this inequality by solving each of the inequalities separately in turn. First, we consider
By subtracting four from each side, we get
We can now divide each side by 7 to find that
Now, we need to solve
We subtract four from each side to get and then divide each side by seven to get
If we combine the two inequalities, we can see that is greater than one but less than or equal to four:
We could represent this solution as an interval, , or on a number line:
Example 2: Solving Compound Inequalities
Find all the values of that satisfy or .
Answer
We can solve this inequality by solving each of the inequalities separately in turn. We look first at
To solve the first inequality, we need to divide each side by five to get
To solve the second inequality, we first need to subtract three from each side to get and then we divide each side by seven:
So, we know that is less than three or greater than or equal to seven. These inequalities can not be combined, so we present our solutions together:
We could also present our solution on a number line:
In the case of inequalities that hold simultaneously, it is sometimes possible to solve them together rather than solve each inequality independently. In the next example, we will demonstrate this method.
Example 3: Solving Compound Inequalities
Find all the values of that satisfy the inequality
Answer
We could separate this into two inequalities and solve each independently. However, it is also possible to solve them together as we will demonstrate here. We start by subtracting three from all parts of the inequality to get
We can then divide each of the three parts by three, which yields
With the approach we used in the previous example, we need to be particularly careful with negative coefficients. This is because, as you will recall, when we divide or multiply an inequality with a negative number, we need to reverse the inequality sign.
Let us have a look at a more formal example where this is the case.
Example 4: Solving Compound Linear Inequalities
Find all values of that satisfy . Write your answer as an interval.
Answer
There are two approaches for solving this compound inequality. The first is to separate it into two inequalities and then solve them individually. Consider first
We add and 18 to each side to get
Now, consider
Here, we add and one to each side to get
Combining these to inequalities, we see that
We can write this in interval notation as follows: .
Alternatively, we could solve this more directly but we would need to be careful when dividing though by . We will demonstrate this now. Consider again the inequality
We subtract four from all three parts to get
Then, we divide all three parts through by . In doing this, however, we need to reverse the inequalities to get
This can then be written as an interval as follows: .
Let us now look at an example where we have variables on each side of the compound inequality.
Example 5: Solving Compound Linear Inequalities with a Variable on Each Side
Find all values of that satisfy . Write your answer as an interval.
Answer
Our first step is to subtract from each part of the inequality to get
We then need to add five throughout the inequality, giving us
If we then write this as an interval, we get .
Before finishing, let us now look at a pair of compound inequalities that contain an intersection of values. When this is the case, a decision needs to be made as to how the final solution is presented.
Example 6: Solving Compound Inequalities with Overlapping Solutions
Find all the values of that satisfy the compound inequality
Answer
To solve the inequality on the left first, we add five to each side:
Then, we divide both sides by seven to find that
To solve the second inequality, we first subtract three from each side to get
Then, we divide each side by six to find that
We can then present the two solutions on a number line:
As the compound inequalities are connected by an “or,” our final solution needs to satisfy at least one of the inequalities. So, our solution is
Were the inequalities connected by an “and,” our final solution would need to satisfy both inequalities, in which case the solution would have been
Key Points
- Compound inequalities give us two or more conditions that hold either separately or simultaneously.
- Generally, we solve compound inequalities by solving each inequality separately and then combining the solutions.
- Compound inequalities can have no solution, a single unique solution, or infinitely many solutions.
