RWM102 Study Guide

Unit 4: Inequalities

4a. Simplify inequalities using addition and multiplication properties 

  • How do you solve inequalities using addition and multiplication properties?

Solving inequalities is almost the same to solving equations, with just one exception, which we will discuss shortly. For inequalities involving addition or subtraction, the same techniques of doing the "opposite" that we learned for equations will still work to solve inequalities. For example, x-3 < 5 can be solved by adding 3 to both sides, giving a final answer of  x < 8.

Similarly, inequalities with the multiplication or division of positive numbers can be solved just as they were with equations. For example, 3x > 12 can be solved by dividing both sides by 3, giving a final answer of x > 4.

The only notable difference is when you are multiplying or dividing an inequality by a negative number. Any time you multiply or divide an inequality by a negative number, you must "flip" the inequality symbol. For example, to solve -3x >12, you must divide by -3, as normal, but when you do, the greater than symbol will "flip" to become a less than symbol, giving a final answer of x < -4.

To review, see One Step Inequalities.


4b. Find and graph on a number line the solution of a given linear inequality with one variable

  • How do you solve linear inequalities?
  • How do you represent the solution to an inequality?

As we previously saw, solving inequalities is mostly the same as equations. Once you have finished solving an inequality, you can leave your answer with the inequality symbol, but sometimes a more visual approach to the solution is required. We can graph the solution to inequalities on a number line. For example, if the final solution to an inequality is x < 3, then we can graph that solution on the number line as such:

In this case, since x is less than 3, we start our solution at 3, with an open circle, since 3 itself is not a solution. Then we shade all the numbers that are less than 3, as those are the solution to the problem. If the final solution includes a or symbol, then the circle should be filled in, not open.

To review, see Plotting Inequalities on a Number Line


4c. Create inequalities in one variable and use them to solve problems 

  • How do you create an inequality to solve a problem?

When creating inequalities, all of the same rules continue from creating equations. The only difference is that instead of saying something like, "is equal to", inequalities will say something like "is less than", or "is more than or equal to".

Remember the inequality symbols and their definitions, and use them appropriately to write an inequality:

< "is less than"

> "is greater than"

 \leq "is less than or equal to"

\geq "is greater than or equal to"

For example, three more than twice a number is less than or equal to 13. Written as an inequality, we get 2x+3 \leq 13. By solving, we find that x \leq 5.

To review, see Multi-step Inequalities with Variables on Both Sides.


Unit 4 Vocabulary

This vocabulary list includes terms listed above that students need to know to successfully complete the final exam for the course.

  • inequalities
  • number line
  • graph
  • inequality symbols