In the last section, you solved inequalities and evaluated possible answers to see if they were reasonable. In this section, you will graph the solutions to the inequalities.

Review the problems from the previous sections and their graphs by clicking on the problem tab.

Use the graphs above to answer the questions that follow.

• Compare and contrast the graphs.

  Interactive popup. Assistance may be required.

  Need a hint?

  List all the things they have in common and all the things that are different.
  Interactive popup. Assistance may be required.

  Check Your Answer

  - They all have a point at one end of the graph.
  - Three have rays that point to the left, and one has a ray that points to the right.
  - If x is greater than or equal to (x ≥) the value, the ray points to the right.
  - If x is less than or equal to the value (x ≤), the ray points to the left.
• Where do the solutions to the inequality lie on the graph?

  Interactive popup. Assistance may be required.

  Need a hint?

  Replay the interactive if necessary, and notice where the graphs start and end.
  Interactive popup. Assistance may be required.

  Check Your Answer

  The solutions to the inequalities are the values included in the point and the ray.
• In the examples, the operations were either "greater than" or "equal to" or "less than" or "equal to." If they were only "greater than" or "less than," how would the graphs differ?

  Interactive popup. Assistance may be required.

  Need a hint?

  Review how the solutions to the above examples are represented on the graph. If the inequality does not include "equal to," decide the value that is not included.
  Interactive popup. Assistance may be required.

  Check Your Answer

  The endpoint on the graph wouldn’t be a solution; therefore, the endpoint circle would not be filled in, which indicates the solution set is everything up to but not including the endpoint on the graph.

Use the interactive below to practice graphing inequalities. Match the inequality with its graph.

Pause and Reflect

• If the inequality is greater than or equal to (x ≥), the endpoint is part of the solution, the point is filled in and the remaining solutions would be to the right of the endpoint.
• If the inequality is greater than (x >), the endpoint is NOT part of the solution, the point is NOT filled in and the remaining solutions would be to the right of the endpoint.
• If the inequality is less than or equal to (x ≤), the endpoint is part of the solution, the point is filled in and the remaining solutions would be to the left of the endpoint.
• If the inequality is less than (x <), the endpoint is NOT part of the solution, the point is NOT filled in and the remaining solutions would be to the left of the endpoint.

Determine if each endpoint below is included in the solution, and if it is filled in or not and if the remaining solutions would be to the right or left of the endpoint.

1. x ≥ 8
2. y < 6
3. x >1,539
4. a ≤ 6.83

Interactive popup. Assistance may be required.

Check Your Answer

1. Yes, filled in, to the right of the endpoint
2. No, not filled in, to the left of the endpoint
3. No, not filled in, to the right of the endpoint
4. Yes, filled in, to the left of the endpoint.

Practice

1. Julio’s team won $100 in the jingle writing contest. They have to deduct any costs of the project before dividing the winnings. Determine the amount each team member could win. Graph your solution.

   Interactive popup. Assistance may be required.

   Need a hint?

   The most they could split between the team members is $100. If they have to pay for the project costs, the amount they would each receive could be smaller than if they didn’t have project costs.
   Interactive popup. Assistance may be required.

   Check Your Answer

   Let w = the possible amount won for each team member.
   
   8w ≤ 100
   8w ≤ 100
   8          8
   w ≤ 12.50
   
   Each member of Julio’s team will win less than or equal to $12.50.
   
   
2. Jan’s math class is playing an angle measure match game. She has to find someone with an angle measure that would be a complement to her angle or less. Jan selected an angle with a measure of 37°. All the angles have positive measures. Determine the value of the angles Jan could make a match with, and then graph the solution.

   Interactive popup. Assistance may be required.

   Need a hint?

   Complementary angles are angles that, when their measures are added together, have a sum equal to 90°. Jan needs to find an angle that when added to 37° is less than or equal to 90°.
   Interactive popup. Assistance may be required.

   Check Your Answer

   Let x = the measure of an angle Jan is looking for.
   
   37 + x ≤ 90
   37 + x ≤ 90
   -37        -37
   0 + x ≤ 53
   x ≤ 53
   
   Jan needs to find someone with an angle less than or equal to 53°.
   
   
3. Four times a number is greater than 7. Find the possible numbers and graph the solution.

   Interactive popup. Assistance may be required.

   Need a hint?

   Four times a number is the same as 4n. This number is greater than 7.
   Interactive popup. Assistance may be required.

   Check Your Answer

   Let n = the number.
   
   4n >  7
   4n >  7
   4       4
   n > 7 over 4 7 4
   
   The number is greater than 7 over 4 7 4 .