Questions About Inequalities (5 Key Things To Know)


Inequalities are used to compare numbers and to indicate a range of values.  They can also be used to express uncertainty (as in the case of an estimate), but there is a lot more to know about inequalities.

So, what do you need to know about inequalities?  Inequality symbols compare two expressions to tell which one is greater. An inequality symbol can have an equal sign, which tells us that the two sides may be equal. We flip an inequality symbol when we multiply or divide by a negative or take the reciprocal on both sides of an inequality.

Of course, an inequality symbol is another way to express an interval or a set of numbers that are valid values for a variable.

In this article, we’ll talk about inequalities and when to use them.  We’ll also look at some examples of how to solve common inequalities, including those with absolute values.

Let’s get started.

Questions About Inequalities

It helps to know about when and how to use inequality symbols, along with what to remember about flipping them.  We’ll start with inequality symbols with equal signs.

Can An Inequality Have An Equal Sign?

An inequality can have an equal sign, but not all of them do.  We sometimes call an inequality with an equal sign an inclusive (or non-strict) inequality.

The inclusive inequality symbols are <= and >=, and both of these indicate that the endpoint of an interval is included in a set.

The strict inequality symbols are < and >, and both of these indicate that the endpoint of an interval is excluded in a set.

Example 1: An Inequality With An Equal Sign

Consider the inequality 2 <= x <= 5.  Since the inequality symbols have an equal sign, they are both inclusive (non-strict) inequalities.

This means that the two endpoints (x = 2 and x = 5) are both included in the solution set for this inequality.

We can also express this set with square brackets as [2, 5], which is the set of values that is valid for x in this case.

We could also use a number line with solid circles at the endpoints x = 2 and x = 5, with shading between them, to illustrate the same interval.

number line 2 to 5 inclusive
The inequality 2 <= x <= 5 includes the endpoints x = 2 and x = 5.

Example 2: An Inequality Without An Equal Sign

Consider the inequality 2 < x < 5.  Since the inequality symbols do not have an equal sign, they are both strict (exclusive) inequalities.

This means that the two endpoints (x = 2 and x = 5) are both excluded from the solution set for this inequality.

We can also express this set with parentheses as (2, 5), which is the set of values that is valid for x in this case.

We could also use a number line with open circles at the endpoints x = 2 and x = 5, with shading between them, to illustrate the same interval.

number line 2 to 5 exclusive
The inequality 2 <= x <= 5 excludes the endpoints x = 2 and x = 5.

How To Know Which Way The Inequality Sign Goes

When using an inequality sign, there are some ways to help you remember which way the signs go:

  • You can remember that the inequality symbol “points to” the smaller number (that is, the number farther left on the number line).  For example, 5 > 2 is correct, since the inequality symbol is pointing towards 2, which is the smaller number.
  • You can remember that the inequality symbol is an alligator that “eats” the larger number (that is, the number farther right on the number line).  For example, 5 > 2 is correct, since the inequality symbol is “eating” the number 5, which is the larger number.
  • You can remember that “X Less Than Y” uses the symbols “X < Y”, since the inequality symbol “<” looks like the letter “L” tilted sideways (“L” stands for “Less Than”).

What Inequality Sign Is Maximum (At Most Or No More Than)?

If we want to denote a maximum value M for the variable x, we use the notation “x <= M”.  The following phrases and expressions are equivalent:

  • x <= M
  • The maximum value of x is M.
  • The value of x is at most M.
  • The value of x is no more than M.

What Inequality Sign Is Minimum? (At Least Or No Less Than)?

If we want to denote a minimum value m for the variable x, we use the notation “x >= m”.  The following phrases and expressions are equivalent:

  • x >= M
  • The minimum value of x is m.
  • The value of x is at least m.
  • The value of x is no less than m.

When Does An Inequality Sign Flip?

There are several situations when an inequality sign can flip:

  • When you multiply both sides of the inequality by a negative number.
  • When you divide both sides of the inequality by a negative number.
  • When you take the reciprocal of both sides of an inequality (watch out for zero denominators!)
  • When solving some absolute value inequalities
  • When using square roots to solve some inequalities.

Let’s take a look at some examples to see when flipping an inequality symbol comes up.

Example 1: Flipping An Inequality Symbol When Multiplying By A Negative

Let’s say we have the inequality (x – 3) / -2 > 6.  In order to solve, we have to multiply by a negative and flip the inequality symbol:

  • (x – 3) / -2 > 6
  • -2*((x – 3) / -2) < -2*6  [multiply by -2 on both sides, and flip the inequality symbol]
  • x – 3 < -12
  • x < -9

We can pick a value of x that is less than -9 as a sanity check.  We’ll pick x = -11, which is less than -9.

 Substituting x = -11 into the left side of the original inequality, we get:

  • (x – 3) / -2
  • =(-11 – 3) / -2
  • =-14 / -2
  • =7

Since 7 > 6, the value of x = -11 satisfies the original inequality.  So we are more confident in our solution.

Example 2: Flipping An Inequality Symbol When Dividing By A Negative

Let’s say we have the inequality -5x > 30.  In order to solve, we have to divide by a negative and flip the inequality symbol:

  • -5x > 30
  • -5x / -5 < 30 / -5  [divide by -5 on both sides, and flip the inequality symbol]
  • x < -6

We can pick a value of x that is less than -6 as a sanity check.  We’ll pick x = -10, which is less than -6.

 Substituting x = -10 into the left side of the original inequality, we get:

  • -5x
  • =-5(-10)
  • =50

Since 50 > 30, the value of x = -10 satisfies the original inequality.  So we are more confident in our solution.

Example 3: Flipping An Inequality Symbol When Taking the Reciprocal

Let’s say we have the inequality 4 / x > 2 / 41.  In order to solve, we have to take the reciprocal of both sides and flip the inequality symbol:

  • 4 / x > 2 / 41
  • x / 4 < 41 / 2  [take the reciprocal of both sides, and flip the inequality symbol]
  • 4 (x / 4) < 4(41 / 2)  [multiply both sides by 2]
  • x < 164 / 2
  • x < 82

We can pick a value of x that is less than 82 as a sanity check.  We’ll pick x = 41, which is less than 82.

 Substituting x = 41 into the left side of the original inequality, we get:

  • 4 / x
  • 4 / 41

Since 4 / 41 > 2 / 41, the value of x = 41 satisfies the original inequality.  So we are more confident in our solution.

Example 4: Flipping An Inequality Symbol When Solving An Absolute Value Inequality

Let’s say we have the absolute value inequality |3x| > 12.  In order to solve, we have to consider two “halves” of the inequality: the positive and the negative.

For the positive half, we solve 3x > 12, which comes out to x > 4.

For the negative half, we flip the inequality symbol and the sign of the constant and then solve 3x < -12, which comes out to x < -4.

So, the solution set for this inequality is x > 4 or x < -4.

We can pick values of x in both halves of this set as a sanity check.  We’ll pick x = 10 (which is greater than 4) and x = -10 (which is less than -10).

Substituting x = -10 into the left side of the original inequality, we get:

  • |3x|
  • =|3(-10)|
  • =|-30|
  • =30

Since 30 > 12, the value of x = -10 satisfies the original inequality.  So we are more confident in our solution.

Substituting x = 10 into the left side of the original inequality, we get:

  • |3x|
  • =|3(10)|
  • =|30|
  • =30

Since 30 > 12, the value of x = -10 satisfies the original inequality.  So we are more confident in our solution.

Example 5: Flipping An Inequality Symbol When Using Square Roots To Solve

Let’s say we have the inequality x2 > 16.  In order to solve, we have to consider two “halves” of the inequality: the positive and the negative.

For the positive half, we take the square root of both sides and take the positive square root of 16, which comes out to x > 4.

For the negative half, we take the square root of both sides and choose the negative square root of 16 and flip the inequality symbol, which comes out to x < -4.

So, the solution set for this inequality is x > 4 or x < -4.

As before, we can choose values of x in this set as a sanity check.

Why Do You Flip The Inequality Sign When Multiplying Or Dividing By A Negative?

You must flip the inequality sign when multiplying or dividing by a negative because you are changing the sign of the numbers in the solution set.

Remember that if we solve the same inequality with two different methods, we should get the same solution set with both.  Let’s look at an example to illustrate this idea.

Example: Solving An Inequality In Two Different Ways

Consider the inequality -2x < 10.  There are two ways we can solve this inequality.

Our first method will be to use both additive and multiplicative inverses:

  • -2x < 10
  • -2x + 2x < 10 + 2x  [first, add 2x to both sides]
  • 0x < 10 + 2x  [-2x + 2x = 0x]
  • 0 < 10 + 2x  [0x = 0]
  • -10 + 0 < -10 + 10 + 2x  [add -10 to both sides]
  • -10 < 0 + 2x  [-10 + 10 = 0]
  • -10 < 2x
  • -10 / 2 < 2x / 2  [divide by 2 on both sides]
  • -5 < x

So, the solution set is x > -5.

We should get the same answer if we solve it another way: by using only a multiplicative inverse:

  • -2x < 10
  • -2x / -2 > 10 / -2  [divide by -2 on both sides, and flip the inequality symbol]
  • x > -5

We get the same solution with this method, x > -5.  However, this method takes a lot fewer steps, so it helps to remember this rule!

Note: if we did not flip the inequality symbol when dividing both sides by -2, we would have gotten x < -5, which is on the opposite side of the endpoint x = -5.

Conclusion

Now you know more about inequalities and when they are used.  You also know the answers to some common questions about inequalities.

In some cases, an inequality will have no solution, and you can learn more about these cases in my article here.

You can learn more about compound inequalities here.

I hope you found this article helpful.  If so, please share it with someone who can use the information.

Don’t forget to subscribe to my YouTube channel & get updates on new math videos!

~Jonathon

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