Solving Inequalities

Contributor: Mason Smith. Lesson ID: 11240

Whether you notice or not, you use inequalities in different ways. Solving them is possible if you start with one-step linear inequalities. Find out how with online practice and real-world examples!

categories

Algebra I, Expressions and Equations, Pre-Algebra

subject
Math
learning style
Visual
personality style
Lion
Grade Level
Middle School (6-8), High School (9-12)
Lesson Type
Quick Query

Lesson Plan - Get It!

Audio:

Q: Why did the parents think their little variable was sick?

A: The nurse said he had to be isolated!

In the first lesson in this Inequalities series, you learned what an inequality is and represents.

If you missed it or need a review, catch up in the right-hand sidebar under Related Lessons.

Now, it's time for you to learn to solve inequalities to get your final answer.

Remember, an inequality is just like an equation. Anything you do to the left side of the equation, you must also do on the right side, so the statement remains balanced.

For example: If an equation says x + 4 < 9 , for every value of x where the statement is true, the inequality is considered to be true.

The maximum number where the statement would be true is just under five since, 4.9999999 + 4 is 8.9999999 which is < 9.

In theory, we could go out as many decimal places as we'd like (4.999999999....) because, even if the 9s repeat forever, it will never be 5. (This is a difficult idea in math that takes some time to wrap your head around, so no worries if it doesn't quite make sense)

This means x can be all values less than five non-inclusive.

Non-inclusive mean without being the final number, in this case 5.

If we were to have the inequality x ≤ 5, then we would say that x is less than or equal to 5. Another way is to say less than 5 inclusive.

Let's practice solving some inequalities with addition and subtraction:

  • x + 9 < 15

We just subtract 9 from both side, sand we get our answer of x < 6.

  • d - 3 > -6

We just add 3 to both sides and get our answer of d > -3

Sometimes, however, the inequality is slightly more difficult becuase it involves multiplication or division.

  • What values can p have if 2p > 8?

P can be all numbers greater than 4, which we know by dividing 8 by 2 to get p by itself.

  • What values can r have in 2/3 r < 6?

Well, by multiplying by the reciprocal (3/2) we can find r < (18/2) or r < 9.

Okay. That seems simple enough.

However, there is one major exception to solving inequalities where they differ greatly from equations.

With inequalities, if you divide or multiply by a negative, then you have to remember to switch the sign of the inequality.

For example: If we have -2p > 8, when we divide by -2, we have p < -4 instead of p > -4.

Let's do a few examples to make sure we are clear on this idea:

  • Let's take the equation -3 ≤ x/-5

Multiply both sides by -5 to get x by itself.

Then, we end up with -5 * -3 ≥ x or 15 ≥ x.

Note the direction of the inequality throughout the solving process.

  • -4x > 8

When we divide the whole equation by the -4, we get x < -2.

Now that we know how to solve simple inequalities using addition, subtraction, multiplication, and division, let's practice a bit.

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