Inequalities word problems

Question

Ugh! I need help turning these words into an equation. The first two I have no idea but i attempted #'s 3, 4 & %

 

1.  Geometry: The length of a rectangle is 8 feet and its area is less than 168 square feet. Write and solve an inequality to represent the width of the rectangle.

 

 

2. Play ground students at a playground are divided into 5 equal groups with at least 6 students in each group. Write and solve an inequality to represent the number of students at the playground.

 

3. Eight times a number N is less than 27

 

    8 * N= -27 ? 

 

4. A number t divided by 32 is at most 4.25

 

  t/32 > 4.25

 

5. 225 is no less than 12 times a number W

 

225 < 12*W?

Answer 1

1. 8x < 168

      x < 168/8

      x < 21

 

3. 8n < 27

      n < 27/8

      

 

4. t/32 ≤ 4.25

         t ≤ (32) (4.25)

         t ≤ 136

 

5. 225 > 12w

    12w < 225

       w < 225/12

Answer 2

I thought I would add some more explanation to what Jane has provided above.

 

Question 1

You want to use the words in the problem to help you understand what's going on. Looking at the first problem, the key words are LENGTH and AREA. Remember that AREA = LENGTH * WIDTH. From that definition we can build our inequality statement, working symbol by symbol.

 

The quantity associated with the AREA is 168 ft². 168 ft² is bigger than the area of our mystery rectangle so use the greater than (>) sign. The quantity associated with the LENGTH is 8 ft. We don't know the WIDTH; I usually recommend choosing a common sense variable like "w" to represent the width. Now we can write the expression, divide both sides by 8 ft to solve for w:

168 ft² > (8 ft) w 

21 ft > w

 

Question 2

We want to think about what is happening on the playground. We have a total number of students, and they are standing around in 5 groups. There are at least 6 students in each group. Now we can begin to translate it into math. We are DIVIDING the total number of students into 5 groups. Mathematically, we would write "n/5" where n is the total number of students. The "at least" tells us that it is possible for exactly 6 students to be in a group. We're ready to write the expression, multiply both sides by 5 to solve for n.

 

n/5 ≥ 6 students

n ≥ 30 students

 

Question 3

I love that you started to answer this question. I see how you used the "less than" in the problem statement and thought that indicated subtraction. If we were writing equations, you took the exact right approach. This statement though is an inequality so you would want to use the less than sign. We can write the expression below, and divide both sides by 8 to solve for n.

8n < 27

n <27/8

 

Question 4

You have another really good start to this question. The key question to ask yourself is which side is bigger? You can test the logic by asking questions like "Can t/32 be equal to 5?" Since the answer to the test question is no, 4.25 is the bigger side of the expression. Also t/32 CAN be equal to 4.25. That gives us the expression below where we need to multiply both sides by 32 to solve for t.

t/32 ≤ 4.25

t ≤ 136

 

Question 5

You're so close on this question. Sometimes it helps to change the words into a more English friendly statement. Instead of "no less" you can use 225 is not less than 12 times a number W. If it's not less than, that means 225 has to be greater than or equal to. Now we can we can write the expression, divide both sides by 12 to solve for W.

225 ≥ 12W

225/12 ≥ W

 

 

As a final note for all of these inequalities, there is no correct "side" to put the variable on. Some teachers ask students to always put the variable on the left; that is their preference so they can provide more consistent feedback.