WRITING INTERVAL NOTATION GIVEN A NUMBER LINE

Use interval notation to describe :

Problem 1 :

Solution :

By observing the number line, the possible values of x are

-2 ≤ x < 1

By expressing it as interval notation, we get

[-2, 1)

Problem 2 :

Solution :

By observing the number line, the possible values of x are

x ≤ 0 or x > 1

By expressing it as interval notation, we get

(-∞, 0] and (1, ∞)

Problem 3 :

Solution :

By observing the number line, the possible values of x are

2 ≤ x ≤ 4

By expressing it as interval notation, we get

[2, 4]

Problem 4 :

Solution :

By observing the number line, the possible values of x are

-2 < x < 1

By expressing it as interval notation, we get

(-2, 1)

Problem 5 :

Solution :

By observing the number line, the possible values of x are

x < -1 (or) x > 1

By expressing it as interval notation, we get

(-∞, -1) and (1, ∞)

Problem 6 :

Solution :

By observing the number line, the possible values of x are

0 < x ≤ 4

By expressing it as interval notation, we get

(0, 4]

Problem 7 :

Solution :

By observing the number line, the possible values of x are

x < 0 (or) x ≥ 3

By expressing it as interval notation, we get

(-∞, 0) and [3, ∞)

Problem 8 :

Solution :

By observing the number line, the possible values of x are

x ≤ -2 (or) x ≥ 2

By expressing it as interval notation, we get

(-∞, -2] and [2, ∞)

Problem 9 :

Solution :

By observing the number line, the possible values of x are

x < -2 (or) 1 < x ≤ 4

By expressing it as interval notation, we get

(-∞, -2) and (1, 4]

Problem 10 :

Solution :

By observing the number line, the possible values of x are

-2 ≤ x ≤ 1 (or) x > 3

By expressing it as interval notation, we get

[-2, -1] and (3, ∞)

Problem 11 :

The water temperature of a swimming pool must be no less than 76°F. The temperature is currently 74°F. Which graph correctly shows how much the temperature needs to increase? Explain your reasoning.

Solution :

Let x be the temperature.

The temperature is no less than 76°F, then x ≥ 76

Current temperature = 74°F

So, we have to increase atleast 2°F. Option C is correct.

Problem 12 :

According to a state law for vehicles traveling on state roads, the maximum total weight of a vehicle and its contents depends on the number of axles on the vehicle. For each type of vehicle, write and graph an inequality that represents the possible total weights w (in pounds) of the vehicle and its contents.

Solution :

The inequalities that represent the maximum weights are:

• 2 axles - x ≤ 40000
• 3 axles - x ≤ 60000
• 4 axles - x ≤ 80000

Let the maximum weight be represented with x

In inequality, maximum means less than or equal to i.e. ≤

(a) 2 axles

The maximum weight, here is 40000.

So, the inequality is:

x ≤ 40000

(b) 3 axles

The maximum weight, here is 60000.

So, the inequality is:

x ≤ 60000

(c) 4 axles

The maximum weight, here is 80000.

So, the inequality is:

x ≤ 80000

Problem 13 :

The graph represents the known melting points of all metallic elements (in degrees Celsius).

a. Write an inequality represented by the graph.

b. Is it possible for a metallic element to have a melting point of −38.87°C? Explain.

Solution :

a) Let x be the temperature.

Since we have solid circle, we have to use the less than or equal to greater than or equal sign.

x ≥ -38.87

b) Yes, it is possible for a metallic element to have a melting point of  -38.87 because this is the melting point of the metallic element mercury.

Problem 14 :

write an inequality that represents the missing dimension x.

The area is less than 42 square meters.

Solution :

Base = x meter and height = 6 m

Area of triangle = (1/2) • base • height

= (1/2) • x • 6

= 3 x

Area of the triangle is less than 42 square meter.

3x < 42

Solving for x, we get

x < 42/3

x < 14

Problem 15 :

The area is greater than or equal to 8 square feet

Solution :

Base = x ft and height = 10 ft

Area of triangle = (1/2) • base • height

= (1/2) • x • 10

= 5 x

Area of the triangle is greater than 8 square feet

5x > 8

Solving for x, we get

x > 8/5

Problem 16 :

The area is less than 18 square centimeters.

Solution :

Total area = area of triangle + area of rectangle

= (1/2) • 4 • x + 4 • x

= 2x + 4x

= 6x

Area is less than 18 square cm

6x < 18

x < 18/6

x < 3

Problem 17 :

The area is greater than 12 square inches

Solution :

Length = 2 inches and width = x inches

Area of rectangle > 12

2x > 12

x > 12/2

x > 6