In this section, you will write inequalities that represent a real-world problem situation. First, practice with an example.

Source: Cherry Tomato, James Maxon, Public Domain Pictures

Good soil, sunlight, and cool nights are needed for growing tomatoes in Texas. Seeds are usually planted inside around mid-January or the beginning of February. Suzy planted her seeds the first week of February. Her tomato seeds will turn into plants, and then tomatoes will grow on the plants until the end of June. How many weeks will pass from the first week of February through the end of June?

Let w = the number of weeks Suzy’s tomato plants have to produce tomatoes.

(There are approximately 22 weeks from February 1st until June 30th.)

w ≤ 22

Use the interactive below to match each problem situation with an inequality. It is possible that all of the equations are not necessary. To make the match, drag the inequality next to the problem.

Use the interactive to answer the following questions.

• Three of the inequalities are addition problems. If they were equations, how would you solve them?

  Interactive popup. Assistance may be required.

  Need a hint?

  Determine how to move the addend to the opposite side.
  Interactive popup. Assistance may be required.

  Check Your Answer

  Add the opposite of the constant term to both sides.
• There is one inequality that is a multiplication problem. If it was an equation, how would you solve it?

  Interactive popup. Assistance may be required.

  Need a hint?

  The coefficient in front of the variable has to be 1 in order to determine the solution to the equation.
  Interactive popup. Assistance may be required.

  Check Your Answer

  Multiply both sides by the reciprocal or divide by the coefficient.
• Two of the problems have a less than or equal to sign and two of the problems have a greater than or equal to sign. Is it possible to write all the problems with a less than or equal to sign? Explain.

  Interactive popup. Assistance may be required.

  Need a hint?

  In order to use only a less than or equal to sign the information needs to be moved.
  Interactive popup. Assistance may be required.

  Check Your Answer

  Yes. As long as the statement is still true, all of the inequalities could be written with a less than or equal to sign. The information on the left will move to the right and the information on the right will be moved to the left in order to make sure that the inequality is still true.

Practice

1. Charles is saving for a pair of running shoes. He found two pairs that he liked. The cost of one pair is $168, and the cost of the second pair is $159. He has already saved $63. Write an inequality showing the amount Charles needs to save to purchase either pair of the running shoes.

   Interactive popup. Assistance may be required.

   Need a hint?

   If Charles doesn’t care which pair of running shoes he would like to purchase, the least amount he needs to save would be for the cheaper pair. But if he saves more, he could get the more expensive pair.
   Interactive popup. Assistance may be required.

   Check Your Answer

   Let x = the amount Charles needs to save.
   
   63 + x ≥ 159
2. A standard elevator usually holds between 1,000 and 6,000 pounds. Sam entered an elevator that had a weight limit of 3,500 pounds. Before entering, Sam saw that the elevator already held a group of people and a dolly filled with supplies. Sam estimated that the elevator already held a combined weight of 750 pounds. Write an inequality showing the weight, in pounds, that the elevator could still possibly hold.

   Interactive popup. Assistance may be required.

   Need a hint?

   The elevator can hold a maximum of 3,500 pounds. The elevator already contains 750 pounds. The weight available can’t go over the maximum weight but doesn’t have to equal the maximum weight.
   Interactive popup. Assistance may be required.

   Check Your Answer

   Let w = the amount of weight the elevator can still hold.
   
   750 + w ≤ 3,500
3. Mercury is a metallic element with the lowest melting point of -38.87°C. Jack wants to find an element with a melting point that, when doubled, is more than mercury’s melting point. Write an inequality that describes the melting point of the element Jack is looking for.

   Interactive popup. Assistance may be required.

   Need a hint?

   When doubled, the element’s melting point is greater than -38.87°C.
   Interactive popup. Assistance may be required.

   Check Your Answer

   Let m = the element Jack is looking for.
   
   2m ≥ -38.87°C