Solving One-Step Linear Inequalities

Solving One-Step Linear Inequalities

In this video lesson, you will learn how you can solve any kind of linear inequality that only requires one step. Learn when you need to subtract, when you need to add, when you need to multiply, and when you need to divide.

Linear Inequalities

Linear inequalities are linear equations with inequality signs instead of equal signs. They compare two different quantities. In math, the linear inequalities that you will come across will usually involve a variable that you need to solve for. You will find these linear inequalities not just in math, but also in use in the real world. You can ask a medical doctor how tall most people are and he might show you a chart and tell you that most people are less than 7 feet. What he has just told you is a linear inequality in words. If we represented a person's height with x, we can write x < 7 feet to show that his height is less than 7 feet. And there we have our linear inequality. It is linear because our variable doesn't have any exponents, or, in other words, has an exponent of 1 (in math, no exponent means there is an exponent of 1 that is not written). We have our 'less than' inequality sign and we have the other side of our inequality, the 7. What we have here is a complete inequality statement because we know exactly what values our x can take (any number less than 7). However, there will be times where you have a problem such as x + 3 < 10 where you need to solve the problem to find out what our x values are.

Solving by Subtraction

In this first example that we are looking at, we see that our x has a plus 3 attached to it. When we see an addition problem like this, we solve it by subtracting the number that is being added. Remember, if you do one operation to one side of an equation or inequality, then you must do the same to the other side. So for the problem x + 3 < 10, we subtract the 3 from both sides of the inequality to solve. We get x + 3 - 3 < 10 - 3 which turns into x < 7. And there we have our answer.

Do you see what we did? We actually performed the opposite operation that we saw to solve our inequality for our variable. Remember that when we solve for a variable, we want to get our variable by itself.

Solving by Addition

Now, if we had a subtraction problem, we would perform the opposite operation to solve it just like we did for the addition problem. So, if we see a problem such as x - 8 > 10, we would add the 8 to both sides of the problem. What do we get? We get x - 8 + 8 > 10 + 8 which turns into x > 18. Our answer is x > 18, or all numbers greater than 18.

Since this lesson is all about linear inequalities that only require one step to solve, we won't talk about the situation where our x is the one being subtracted. That requires two steps and is covered in another lesson.

Solving by Division

If we had a problem such as 4x > 20, what do you think we need to do? That's right; we need to divide, because division is the opposite of multiplication. So, we divide both sides by 4. We get 4x/4 > 20/4 which becomes x > 5.

There is a special situation that can happen when you have a multiplication problem. This is when your x is being multiplied by a negative number. When this happens, and you divide by the negative number, your inequality will flip. If you have a less than sign, it will become a greater than sign, and vice versa. If you have a greater than or equal to sign, it will become a less than or equal to sign, and vice versa. So, if we had -4x > 20, we divide both sides by -4. -4x/-4 > 20/-4. This becomes x < -5. Notice how our inequality sign has flipped. We began with a greater than sign and we ended with a less than sign. The rule that we are following here is that if we multiply or divide by a negative number, then our inequality sign will flip.

Solving by Multiplication

So, what do you think is needed when we have a division problem? What operation do we need to do to solve this type of problem? That's right; we need to multiply. So, to solve the problem x/6 < 20, we multiply by 6. Multiplying 6 on both sides of the inequality, we get (x/6)*6 < 20*6. This becomes x < 120.

Here again, if our problem has a division by a negative number, then when we multiply by that negative number, we need to remember to flip our inequality sign. So, if our problem was x/-6 < 20, then we would multiply both sides by the -6. We get (x/-6)*-6 < 20*-6. This becomes x > -120. Our inequality sign has flipped. We began with a less than sign and ended with a greater than sign.

Lesson Summary

Let's review what we've learned. Linear inequalities are linear equations with inequality signs instead of equal signs. To solve a linear inequality where a number is being added to our variable, we subtract that number from both sides of the inequality. To solve a linear inequality where a number is being subtracted from our variable, we add that number to both sides of the inequality. To solve a linear inequality where a number is being multiplied with our variable, we divide both sides by that number. If this number is negative, then we make sure to flip our inequality sign. To solve a linear inequality where our variable is being divided by a number, we multiply both sides by that number. If this number is negative, then we make sure to flip our inequality sign just like we do for our multiplication problem when we divide by a negative number.