GED Mathematical Reasoning: Inequalities I

  • An inequality looks like an equation, but instead of an equal sign it contains an inequality sign. The four basic inequality signs are: the “greater than” sign, the “less than” sign, the “greater than or equal to” sign and the “less than or equal to” sign.

> “greater than”
< “less than”
\geq “greater than or equal to”
\leq “less than or equal to”

  • You may be happy to know that solving an inequality involves pretty much the same process as solving an equation, except for one small twist: If you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.
  • Why do we flip the inequality sign when we multiply or divide by a negative number?It all goes back to the nature of negative numbers and the fact that the larger the number part of a negative number is, the smaller the number actually is. Take for example the TRUE statement:  20 greater than 10

    What happens when we divide both sides of this inequality by -2?

    20 divided by -2 is equal to -10

    10 divided by -2 is equal to -5

    The resulting statement is:  -10 greater than -5

    20 \div -2 > 10 \div -2
    -10 > -5

    However notice that this statement isn’t TRUE unless we flip the inequality sign. -10 is NOT greater than -5, it’s less than -5.

    Flip the inequality sign
    -10 < -5 is TRUE

    And the same scenario would be true if we had multiplied both sides of the inequality by a negative number. The resulting statement wouldn’t be true unless we flipped the inequality sign.

  • In general, an open dot is used on the graph when the inequality sign is “greater than” or “less than” and a closed dot is used when the inequality sign is “greater than or equal to” or “less than or equal to.”

< or > \, \rightarrow open dot

\leq or \geq \, \rightarrow closed dot

x < -5  algeb59
x > 2  algeb60
x \leq -5  algeb61
x \geq 2  algeb62

 

Example 1

Solve: -2x + 1 > 11

 

Notice that the statement given in example one looks a lot like an equation, but instead of an equal sign we see a “greater than” sign. We will solve this inequality like an equation and all these things still hold true:

First, solving an inequality means to find the values that make the inequality true. These values make up the solution “set.”

Also, just like with an equation, to solve an inequality we’ll use the concept of opposite operations to get the variable by itself on one side of the inequality sign.

It is still a good idea to check the solution to make sure the result is a true statement.

To isolate the variable “x” on the left side, we will need two steps involving opposite operations. We first need to subtract 1 from both sides. And then as a second step, we will divide both sides by -2.

-2x + 1 - 1> 11 - 1
-2x > 10

Now we are left with the one-step inequality: -2x greater than 10. To solve for x, we will divide both sides by -2.

On the left side, -2 divided by -2 equals one, leaving just “x” on the left side.

On the right side, we need to divide 10 by -2, which is equal to -5.

And since we divided both sides BY a negative number, we need to flip the inequality sign to “less than.”

algeb58

The resulting solution is:  “x” less than -5.

The solution: “x” less than -5 means that any number less than -5 will make the original inequality true. Let’s check this to be sure. Pick any number less than -5 and substitute it back into the original inequality for “x.” I’ll choose “x” equal to -7. When we let “x” equal -7, we get the statement:  -2 times -7 plus 1 is greater than 11. Is this true?

Check: let x = -7
-2x + 1 > 11
-2(-7) + 1 > 11
14 + 1 > 11
15 > 11 is TRUE

Since 15 is, in fact, greater than 11, yes! The statement is true and we can be confident in our solution.

Another thing you should know is that we can represent the solution set of an inequality visually using a number line. We can draw a picture of the solution set “x” less than -5 by shading all the numbers less than -5 on a number line.

algeb59

Notice that I used an open dot ON the number -5. This is to denote that the number -5 is NOT included in the solution set, since it does not make the inequality true.

Let’s try it. If we let “x” equal -5, is -2 times -5 plus 1 greater than 11?

No, the left side simplifies to 11 and 11 is not greater than 11. So the statement is false, meaning that the number -5 should not be included in the solution set.

Let: x = -5
-2x +1 > 11
-2(-5) + 1 > 11
10 + 1 > 11
11 > 11 is FALSE

 

Example 2

Solve: -4(n + 5) + 7n \leq 10

 

As we consider solving the inequality in example 2, let’s review the steps for solving multi-step equations. Remember – these are the same steps we need to use for solving inequalities, except for that additional step of flipping the inequality sign if we multiply or divide by a negative number.

  • First, simplify each side of the equal sign separately, if possible, by distributing to get rid of any parenthesis and combining like terms.
  • Second, add and subtract to get the variable terms on one side of the equation and the constant terms on the other side.
  • Third, multiply or divide to get the variable by itself on one side of the equal sign.
  • Finally, check your solution.

The inequality in example 2 contains parenthesis so let’s first distribute the -4 on left side.

-4(n + 5) + 7n \leq 10
-4n +  -20 + 7n \leq 10
-4n - 20 + 7n \leq 10

Next, let’s combine the “like terms”

-4n - 20 + 7n \leq 10
3n - 20 \leq 10

Now that we’ve completed step one of the process by distributing and simplifying what we can, let’s move on to step 2 and solve for “n” by first adding 20 to both sides. This will put the variable term on the left side of the inequality and the constant terms on the right side.

algeb63

Now we are left with a one-step inequality: 3n less than or equal to 30.

Following step 3 of the process, we will solve for “n” by dividing both sides by 3.

3n \leq 30

n \leq 10

We did not divide by a negative number, so we will not flip the inequality sign. The resulting solution is:  “n” less than or equal to 10

Visually, this solution set looks like this, where the numbers less than 10 are shaded and there’s a closed dot on the number 10 to denote that the number 10 IS included in the solution set.

algeb65

We can check our solution by choosing a number in the shaded region and substituting it into our original inequality. Let’s choose “n” equal to zero. When we simplify the left side according to the order of operations, is it less than or equal to 10?

Check: n = 0

-4(n + 5) + 7n \leq 10
-4(0 + 5) + 7(0) \leq 10
-4(5) + 0 \leq 10
-20 + 0 \leq 10
-20 \leq 10 is TRUE 

Yes! The resulting statement is: -20 less than or equal to 10, which is a true statement.

As a final note, I would like to show you why we include the endpoint of 10 in the solution set. Let’s look at what happens when we substitute “n” equals 10 back into the original inequality.

Check: n = 10

-4(n + 5) + 7n \leq 10
-4(10 + 5) + 7(10) \leq 10
-4(15) + 70 \leq 10
-60 + 70 \leq 10
10 \leq 10 is TRUE

When we simplify the left side according to the order of operations, we get:  10 less than or equal to 10. While 10 is not less than 10, it IS equal to 10. Therefore, the statement is true and the number 10 is included in the solution set.

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