Solving Inequalities Using All 4 Basic Operations

Hello, and welcome to this video on solving inequalities. In this video, we will discuss:

• What an inequality is, and
• How to solve inequalities using addition, subtraction, multiplication, and division

When solving equations, you have two expressions that are equal to each other. When we look at inequalities, we are looking at two expressions that are “inequal” or unequal to each other, as the name suggests. This means that one equation will be larger than the other. The four basic inequalities are: less than, greater than, less than or equal to, and greater than or equal to.

 Less than < Less than or equal to ≤ Greater than > Greater than or equal to ≥

When solving inequalities, you follow all the same steps as solving an equation, except for a special rule when it comes to multiplication and division. The main difference is that instead of writing an equal sign between the two expressions, you will write one of the four inequality symbols.

Let’s first look at an inequality using addition.

x + 7 is greater than or equal to 4

x + 7 ≥ 4

If we are solving for x by itself, we want to get rid of that 7 next to it, so we subtract 7 from both sides.

x + 7 – 7 ≥ 4 – 7

x is greater than or equal to negative 3

x ≥ -3

It’s as simple as that!

Now, I want you to try one on your own using subtraction.

x minus 3 is less than 9

x – 3 < 9 First, we are going to add 3 to both sides. x – 3 + 3 < 9 + 3 Then we simplify. x < 12 Now we come to multiplication and division. Are you ready to find out what this special rule is that I was talking about earlier? When you multiply or divide by a negative number, you have to flip your sign the opposite direction. If you are multiplying or dividing by a positive number, don’t worry about this step. Let’s look at an example: negative 4x is greater than 12 -4x > 12

To get x by itself, we need to divide both sides by negative 4.

Remember, since we are dividing by -4, we have to flip our inequality sign! So x is less than negative 3.

x < -3 Let’s take a second to look at why this happens. What if I didn’t flip my sign? I would have x is greater than negative 3 x > -3. So let’s try plugging in 2, since 2 is greater than negative 3. If we plug in 2 for x, we get negative 4 times 2 is greater than 12. Negative 8 is greater than 12.

-4(2) > 12
-8 > 12

But we know that this isn’t true. Negative 8 is not greater than 12.

Now look back at our correct answer, x is less than negative 3. Negative 20 is less than negative 3, so let’s plug this into our equation to check and see if it works.

Negative 4 times negative 20 is greater than 12
80 is greater than 12

-4(-20) > 12
80 > 12

That’s true! 80 is greater than 12. So just remember, when you multiply or divide by a negative number, you HAVE to flip the sign. Otherwise, your inequality will not be true.

What if we had this inequality?

x/3 ≤ 2
x over 3 is less than or equal to 2

For this inequality, we need to multiply both sides by 3. When we do this, do we flip our sign? No, we don’t have to since we are multiplying by a positive number.

So we’ll multiply both sides by 3, then we get x is less than or equal to 6.
x ≤ 6

I want you to try one more on your own. For this one, we are going to combine everything we’ve learned, so it will look a little more challenging, but you can do it. Just apply each step that we have talked about so far.

2x plus 3 is greater than or equal to x minus 7
2x + 3 ≥ x – 7

Pause this video and solve this inequality on your own, then see if your answer matches up with mine.

Think you’ve got it? Let’s see!

First, I’m going to add 7 to both sides of my equation.

This gives us 2x plus 10 is greater than or equal to x
2x + 10 ≥ x

Now, I have to subtract 2x from both sides.

10 is greater than or equal to negative x 10 ≥ -x

Finally, I need to divide by negative 1 and flip my sign.

So our final answer is negative 10 is less than or equal to x

-10 ≤ x

Now, notice with this inequality, you could have subtracted x and subtracted 3 from both sides. This will give you the same answer, and you can avoid dividing by a negative. Sometimes there are multiple ways to solve an inequality or an equation, so be on the lookout for ways to make your life a little bit easier.

I hope this video on solving inequalities was helpful. Thanks for watching and happy studying!

Practice Questions

Question #1:

Solve the following inequality for x:
$$4+x$$ < $$-1-x$$

$$x>\frac{1}{2}$$

$$x$$ < $$2$$

$$x$$ < $$-\frac{5}{2}$$

$$x>\frac{1}{4}$$

To solve this inequality, remember that we need to get x by itself on one side, just like with regular equations. First, move the 4 to the right side by subtracting 4 from both sides.
$$4-4+x$$ < $$-1-x-4$$

Simplifying, this leaves us with:
$$x$$< $$-5-x$$

Now, we move that negative x to the left side by adding it to both sides.
$$x+x$$< $$-5$$
$$2x$$ < $$-5$$

Finally, we will divide by 2 on both sides to arrive at the solution. The inequality $$4+x$$ < $$-1–x$$ is satisfied whenever $$x$$ < $$-\frac{5}{2}$$.

Question #2:

Solve the following inequality for x:
$$\frac{x}{4}\geq3$$

$$x\geq12$$
$$x\geq4$$
$$x\leq12$$
$$x\leq3$$

To see this, we just need to multiply both sides by 4. This will eliminate the denominator on the left and leave us with x by itself:
$$4\times\frac{x}{4}\geq3\times4$$

Simplify:
$$x\geq12$$

Question #3:

Solve the following inequality to determine which values of x will satisfy it:
$$\frac{x+2}{7} > 2-x$$

$$x>5$$
$$x>\frac{3}{2}$$

$$x$$ < $$\frac{2}{7}$$

$$x$$ < $$2$$

To begin solving this inequality, let’s eliminate the denominator on the left by multiplying both sides by 7.
$$7\times\frac{x+2}{7}>(2-x)\times7$$

Notice that we have not flipped the inequality sign. That’s because we multiplied by a positive number. The above expression simplifies to:
$$x+2>14-7x$$

From here, we will move the 2 to the right side with subtraction, and move the $$7x$$ to the left side with addition.
$$x+2-2>14-7x-2$$
$$x>12-7x$$
$$x+7x>12-7x+7x$$
$$8x>12$$

Finally, we divide both sides by 8 to get x by itself and we see that we are left with $$x>128$$, which simplifies to $$x>32$$.

Question #4:

Solve the following inequality for x.
$$-4x+2\geq6$$

$$x\leq-1$$
$$x\geq-1$$
$$x\leq2$$
$$x\geq2$$

Let’s begin working on this problem by subtracting 2 from both sides.
$$-4x+2-2\geq6-2$$
$$-4x\geq4$$

Now, in order to isolate x and find the solution, we need to divide both sides by -4. Remember, dividing by a negative will flip the inequality sign! This is also true of multiplication by negatives.
$$\frac{-4x}{-4}\geq\frac{4}{-4}$$
$$x\leq-1$$

Question #5:

Solve the following inequality for x.
$$-\frac{1}{4}x+3>-4$$

$$x$$ < $$16$$

$$x$$ < $$28$$

$$x>34$$
$$x>21$$

First, move the 3 to the right-hand side of the inequality by subtracting 3 from both sides.
$$-\frac{1}{4}x+3-3$$>$$-4-3$$
$$-\frac{1}{4}x$$>$$-7$$

Now, we will get x by itself by multiplying both sides by -4. This will cancel the fraction on the left-hand side. This will cause the inequality sign to flip and leave us with our solution after we simplify.
$$(-4)×-\frac{1}{4}x$$>$$-7×(-4)$$
$$x$$ < $$28$$