PROBLEMS ON GRAPHING INEQUALITIES ON THE NUMBER LINE

There are four inequality signs,

While representing linear inequalities in one variable on number line , we may use two circles.

(i) Transparent circle (or) Empty circle (or) Unfilled circle

(ii) Solid circle (or) Filled circle

In the question,

• If we have < or > signs, we have to use transparent circle to represent the solution.
• If we have ≤ or ≥ signs, we have to use the solid circle to represent solution.

Solve the inequalities and represent the possible values of the variable on a number line.

Problem 1 :

6 > z - 2

Solution :

6 > z – 2

Add 2 on both sides.

6 + 2 > z – 2 + 2

z < 8

Problem 2 :

g + 7 < -12

Solution :

g + 7 < -12

Subtract 7 on both sides.

g + 7 - 7 < -12 – 7

g < -19

Problem 3 :

d – 5 < 7

Solution :

d – 5 < 7

Add 5 on both sides.

d – 5 + 5 < 7 + 5

d < 12

Problem 4 :

15 > k + 2

Solution :

15 > k + 2

Subtract 2 on both sides.

15 - 2 > k + 2 – 2

k < 13

Problem 5 :

1 + x > -16

Solution :

1 + x > -16

Subtract 1 on both sides.

1 - 1 + x > -16 – 1

x > -17

Problem 6 :

y + 8 < -9

Solution :

y + 8 < -9

Subtract 8  on both sides.

y + 8 - 8 < -9 – 8

y < -17

Problem 7 :

8 ≤ 8 + r

Solution :

8 ≤ 8 + r

Subtract 8 on both sides.

8 – 8 ≤ 8 – 8 + r

r ≥ 0

Problem 8 :

w + 8 ≥ 11

Solution :

w + 8 ≥ 11

Subtract 8 on both sides.

w + 8 - 8 ≥ 11 – 8

w ≥ 3

Problem 9 :

The NASA Solar Probe can withstand temperatures up to and including 2600°F. Write and graph an inequality that represents the temperatures the probe can withstand.

Solution :

temperatures t up to and including 2600°F

Let t be the temperatures the probe can withstand.

An inequality is t ≤ 2600

Problem 10 :

Describe and correct the error in graphing the inequality.

Solution :

By observing the graph, the values of x is greater than or equal to 1. The circle should be closed circle.

Problem 11 :

Describe and correct the error in graphing the inequality.

Solution :

By observing the graph, the values of x is lesser than to 1. The circle should be open circle.

Problem 12 :

Each day at lunchtime, at least 53 people buy food from a food truck. Write an inequality that represents this situation.

Solution :

Let x be the number of people buy food for food truck.

x ≥ 53 (at least)

Problem 13 :

A subway ride for a student costs $1.25. A monthly pass costs $35.

a. Write an inequality that represents the number of times you must ride the subway for the monthly pass to be a better deal.

b. You ride the subway about 45 times per month. Should you buy the monthly pass? Explain.

Solution :

a) Let x number of times you must ride the subway.

1.25 x < 35

x < 35/1.25

x < 28

So, the minimum number of times the monthly pass t be used is 28.

b) When x = 45

= 1.25 (45)

= 56.25

Since the amount what we receive is greater than 35, we should purchase monthly pass to reduce expenses.

Problem 14 :

Consider the inequality b > − 2.

a. Describe the values of b that are solutions of the inequality.

b. Describe the values of b that are not solutions of the inequality. Write an inequality for these values.

c. What do all the values in parts (a) and (b) represent? Is this true for any inequality?

Solution :

a) Given that, b > − 2

The values of b should be greater than -2.

b) The values of b should be lesser than -2 including -2.

c) These values represent all real numbers. This is true for any inequality because every real number is either a solution or it is not.

Problem 15 :

A postal service says that a rectangular package can have a maximum combined length and girth of 108 inches. The girth of a package is the distance around the perimeter of a face that does not include the length.

a. Write an inequality that represents the allowable dimensions for the package.

b. Find three different sets of allowable dimensions that are reasonable for the package. Find the volume of each package.

Solution :

a)

l + w + h + w + h ≤ 108

Inequality Representation

l + 2w + 2h ≤ 108

b) Reasonable Package Dimensions

Dimensions: L = 40 inches, W = 10 inches, H = 12 inches

Volume: 40 * 10 * 12 = 4800 cubic inches

Dimensions: L = 50 inches, W = 10 inches, H = 9 inches

Volume: 50 * 10 * 9 = 4500 cubic inches

Dimensions: L = 60 inches, W = 8 inches, H = 7.5 inches

Volume: 60 * 8 * 7.5 = 3600 cubic inches