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The Difference Between Solving An Equation And Solving An Inequality.

Solving Inequalities is very similar to solving Equations.

 

The same general technique applies.

 

That technique for solving equations is:

 

Whatever you do to one side of the equation, you have to do to the other side to preserve the equality

 

The technique for solving inequalities is:

 

Whatever you do to one side of the inequality, you have to do to the other side to preserve the inequality.

 

the techniques are the same.

 

The technique means:

 

If you add or subtract a number to one side of an equation or inequality, you have to add or subtract the same number to the other side of the equation or inequality.

 

If you multiply or divide one side of an equation or inequality by the same number, you have to multiply or divide the other side of the equation or inequality by the same number.

 

The difference between solving equations and solving inequalities is: 

 

If you multiply or divide an inequality by a negative number, then the inequality reverses. !!!!!

 

The following examples will show this rule in action.

 

The first example will be multiplication.

 

The second example will be division.

 

The first example starts with:

 

7 is greater than 5.

 

Multiply both sides of that inequality by -5 and you get:

 

7*-5 = -35 on the left side of the inequality.

 

5*-5 = -25 on the right side of the inequality.

 

The result is -35 is smaller than -25.

 

The inequality started as greater than and became smaller than because you were multiplying both sides of the inequality by a negative number.

 

It’s clear to see that 7 really is greater than 5.

 

It is also clear to see that -35 is less than -25.

 

The second example is simply the reverse of the first example and starts with:

 

-35 is smaller than -25.

 

divide both sides of this inequality by -5 and you get:

 

-35 / -5 = 7 on the left side of the inequality.

 

-25 / -5 = 5 on the right side of the inequality.

 

The result is 7 is greater than 5.

 

The inequality started as less than and became greater than because you were dividing both sides of the inequality by a negative number.

 

You needed to do that to preserve the inequality.

 

It’s clear with these numbers that reversing the inequality is essential when you are mutliplying both sides of the inequality or dividing both sides of the inequality by a negative number.

 

Failure to do this will get you the wrong answer as you will find out when you confirm the results of your solution.

 

To show you what that means, we’ll solve an equality and an inequality using the rules of equation solving and inequality solving.

 

First an equation.

 

-8x = 72

 

To solve this, we want to divide both sides of this equation by -8.

 

We get:

 

-8x / -8 = 72 / -8

 

We simplify this to get:

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x = -9

 

How do we know the operation was successfully completed?

 

We know that because, when we replace x in the original equation with the solution,  the original equation is a true statement.

 

The original equation is:

 

-8x = 72

 

We replace x with -9 to get:

 

-8 * -9 = 72

 

We simplify to get:

 

72 = 72

 

This is a true statement, so we can assume our solution is good.

 

Now we will solve the same problem as an inequality.

 

-8x >= 72

 

This means -8x is greater than or equal to 72.

 

To solve this, we want to divide both sides of this equation by -8.

 

We get:

 

-8x / -8 <= 72 / -8 

 

The inequality had to be reversed because we were dividing both sides of the equation by a negative number !!!!!

 

We started with -8x greater than or equal to 72 and we ended with -8x / -8 smaller than or equal to 72 / -8

 

We simplify this to get:

 

x <= -9

 

How do we know the operation was successfully completed?

 

We know that because, when we replace x in the original equation with the solution, the original equation becomes a true statement.

 

The solution, in this case, is not just one number, but many numbers.

 

The solution is x <= -9.

 

This means x can be equal to -9, or -10, or -11, etc.

 

as long as x is <= -9, we have a solution.

 

To test this inequality, we need to do two things.

 

First is to see if the inequality is false if we violate the rules of the solution.

 

To do that, we select a number greater than -9 and use it to solve the inequality.

 

We’ll try x = 0 because it’s simple and it violates the rules of the solution because 0 is not less than -9.

 

The equation of -8x >= 72 becomes -8 * 0 >= 72 which becomes 0 >= 72 which is a false statement because 0 is not greater than or equal to 72.

 

Since the inequality is false when we replace x with 0, then the solution of x = 0 is not good.

 

This is what we expect since 0 is not less than -9.

 

Next we want to try the conditions where the solution does follow the rules.

 

In this case, we expect the equation to be true.

 

The rules of the solution are that x <= -9

 

We select x = -9

 

The original inequality of -8x >= 72 becomes -8 * -9 >= 72 which becomes 72 >= 72 which is true because 72 is equal to 72.

 

This confirms that, when x = -9, the original inequality is true, so the solution is confirmed as good.

 

We then select x < -9

 

We’ll pick -10 because -10 is less than -9 and it’s easy to calculate.

 

The original inequality of -8x >= 72 becomes -8 * -10 >= 72 which becomes 80 >= 72 which is true because 80 is greater than 72.

 

This confirms that, when x = -10, the original inequality is true, so the solution is confirmed as good.

 

We tested three conditiions for this inequality.

 

x > -9 (false)

 

x = -9 (true)

 

x < -9 (true)

 

The original equation of -8x >= 72 was false when x > -9 as we expected..

 

The original equation of -8x >= 72 was true when x = -9 as we expected.

 

The original equation of -8x >= 72 was true when x < -9 as we expected.

 

Our solution of x <= -9 was confirmed as good.

 

When we say the inequality is preserved, we mean the original inequality is preserved.

 

The original inequality was -8x >= 72

 

In order to preserve that inequality we had to follow the rules of solving inequalities.

 

In the process of solving, the intermediate inequality was reversed, but the original inequality was still  preserved.

 

If you add or subtract the same positive or negative number to both sides of an inequality or an equation, the inequality stays the same.

 

If you multiply or divide both sides of an inequality or an equation by the same positive number, the inequality stays the same.

 

The only difference is:

 

If you multiply or divide both sides of an equation by the same negative number, the equation remains the same, but If you multiply or divide both sides of an inequality by the same negative number, the inequality reverses. !!!!!

 

Keep that in mind and you will have a lot less trouble solving inequalities.

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