The video below will introduce you to inequalities and show you how they are used on the GMAT Exam. Following the video, we also provide the video transcript for your convenience.
Inequalities on the GMAT Exam [Video Transcrip]
Up to this point in the module, we have examined equations where one side of the equal sign has the same value as the other side. In this lesson, we will examine inequalities where the two sides are not equal. The inequality shown here can be read as 6 is less than 7. Another symbol you should be familiar with is the symbol for less than or equal to. So this example can be read as 6 is less than or equal to 7. We can also use this symbol in this way, where we are saying that 7 is less than or equal to 7. We can also express inequalities with this symbol. This is read as 5 is greater than 3. This symbol denoted greater than or equal to as in 5 is greater than or equal to 3.
Now it’s important to note that we can express the same inequality in two ways. For example, we can take the statement 5 is greater than 2 and rewrite it as 2 is less than 5. Both statements here express the same inequality. Similarly, the statements 6 is less than or equal to 7 and 7 is greater than or equal to 6 express the same idea. Now I happen to be partial to taking inequalities and rewriting them so that the inequality sign opens to the right. This way the smaller number is always to the left of the larger number, in the same way, that they appear on the number line.
Okay, now that we have taken care of our general notation, our goal here will be to solve inequalities such as this one. Before we can do this, however, let’s first make some observations regarding the ways in which inequalities behave. Take the inequality 6 is less than 7. You will find that this inequality behaves similar to the way in which equations behave. Here’s what I mean. Let’s take this inequality and subtract 3 from both sides. When we do this we get 3 is less than 4. Since 3 is less than 4, we can see that the inequality remains intact when we subtract 3 from both sides of the inequality. Similarly, if we multiply both sides by 4, the inequality remains since 12 is less than 16. Adding 8 to both sides results in 20 is less than 24, which is a valid inequality. Dividing both sides by 2 results in a valid inequality as well. So it would appear that inequalities behave the same way that equations do, in that if you perform the same operations to both sides, the inequality remains intact.
However, notice what happens when we divide both sides of the inequality by -2. When we do this we get -5 is less than -6, which is false. -5 is not less than -6. So the inequality here does not remain intact when we divide both sides by -2. In fact, it appears that we need to reverse the direction of the inequality here. So from the observations, we can make some general conclusions about inequalities. First, adding and subtracting the same number to or from both sides of the equality does not affect the inequality. Second, multiplying and dividing both sides of the inequality by a positive number does not affect the inequality. And finally, multiplying and dividing both sides by a negative number reverses the inequality. We can now use these results to help us solve inequalities. Essentially if we perform the same operation on both sides of the inequality, the inequality will remain intact. However, if we multiply or divide both sides by a negative number, we must reverse the direction of the inequality.
Okay, let’s try an example. Solving this inequality means finding all values for x that makes the inequality a true statement. So to do this, we need to isolate the variable x. We’ll do this by first subtracting 5 from both sides to get 2x is less than 6. And then we’ll divide both sides by 2 to get x is less than 3. This means the solution to the inequality 2x plus 5 is less than 11, consists of any value of x that is less than 3. As you can see there are infinitely many solutions to this inequality. Now another way to show the solution is to use a number line. To show the solution x is less than 3, we’ll first draw a circle around 3 to denote that 3 is not one of the solutions. Then to represent less than 3, we’ll draw an arrow extending without end to the left. So any number lying on the blue line is a solution to the inequality 2x plus 5 is less than 11.
Okay, here is another example. To solve this inequality, we’ll first subtract 2 from both sides to get -3x is less than or equal to -3. From here we can isolate x by dividing both sides by -3. Now since we are dividing both sides by a negative number, we must be sure to reverse the direction of the inequality. So the solution to the inequality consists of any value of x that is greater than or equal to 1. To show this solution on the number line, we’ll add a dot at 1 to show that 1 is one of the solutions to the inequality. Next to represent greater than 1, we’ll draw an arrow to the right. So any number on the blue line is a solution to the original inequality. To solve this next inequality, we’ll first expand the left-hand side and then simplify the right-hand side. Then we’ll simplify the left-hand side. And then we’ll add 3x to both sides. And from here we can subtract 13 from both sides and then divide both sides by 4 to get -2 is greater than x. Or we can say that x is less than -2. To show this on the number line, we’ll draw a circle around -2 and then draw an arrow to the left. So any point on the blue line here will be a solution to the original inequality.
Now sometimes, you will encounter inequalities with 3 parts. These compound inequalities can be solved using the same techniques we have examined so far. To isolate x in this example, we’ll first subtract 3 from all 3 parts. So 11 minus 3 equals 8. When we subtract 3 from 3 minus 2x we get -2x, and when we subtract 3 from 1 we get -2. From here we can isolate x by dividing all 3 parts by -2. So 8 divided by -2 is -4. Next, since we are dividing all 3 parts by a negative number, we must reverse the direction of this inequality. From here -2x divided by -2 equals x. Once again, we will reverse this inequality. And finally -2 divided by -2 equals 1. To show this on our number line, we will examine the inequality in 2 parts. First we have -4 is less than x. This is the same as x is greater than -4. To show this on the number line, we’ll draw a circle around -4 and then draw an arrow to the right. Next we have x is less than or equal to 1. To show this, we’ll add a dot at 1 and then draw an arrow to the left. So the entire set of solutions for the compound inequality consists of all values of x that are greater than are greater than -4 and less than or equal to 1. Okay, let’s summarize. In this lesson, we learned that solving inequalities is pretty much the same as solving equations. The primary difference is that whenever you multiply or divide both sides by a negative number, you must reverse the inequality.