Inequality Plotting

x > 5

x ≥ 5

Basics
An algebraic inequality can be plotted along a number line.

Sometimes, in order to plot an inequality, you will need to solve an equation.

Question
For x > 5, plot points for whole numbers along the number line below.

Answer
x > 5

Answer Process
Because x > 5, plotting excludes 5, starts with 6, and proceeds higher from there.

Question
For x ≥ 5, plot points for whole numbers along the number line below.

Answer
x ≥ 5

Answer Process
Because x ≥ 5, plotting includes 5 and proceeds higher from there.

Question
For x < 4, plot points for whole numbers along the number line below.

Answer
x < 4

Answer Process
Because x < 4, plotting excludes 4, starts with 3, and proceeds lower from there.

Question
For x ≤ 4, plot points for whole numbers along the number line below.

Answer
x ≤ 4

Answer Process
Because x ≤ 4, plotting includes 4 and proceeds lower from there.

Question
A souvenir shop must generate $160.00 in order to break even. It spends $100.00 on supplies. The shop will charge $12.00 for every souvenir x it sells.

The inequality 12x + 100 ≥ 160 expresses the number of souvenirs the shop must sell in order to break even or make a profit.

Plot the points for which the shop will break even or make a profit along the number line below.

Answer
x ≥ 5

Answer Process
See Equations.

12x + 100 ≥ 160

12x + 100 – 100 = 160 – 100
• 100 is subtracted from each side of = sign
• subtraction is the opposite of addition

12x = 60
• 100 – 100 is the same as 0, so it disappears
• 160 – 100 is the same as 60

$\bf\displaystyle\frac{12x}{12}$ = $\bf\displaystyle\frac{60}{12}$
• each side of = sign is divided by 12
• division is the opposite of multiplication

x = 5
• $\bf\displaystyle\frac{12x}{12}$ is the same as 1x which is the same as x
• $\bf\displaystyle\frac{60}{12}$ is the same as 5
• x has been isolated on the left side of = sign

x ≥ 5
Because the inequality 12x + 100 ≥ 160 specifies greater than or equal to, points plotted along the number line include 5 and proceed higher from there.

Input	Display	Comment
	blinker	clears screen

160 – 100	160-100	subtraction on right side of =

	60

$fraction$ 60 12	$\bf\displaystyle\frac{60}{12}$	division on right side of =

	5

		Inequality

Question
A souvenir shop must generate greater than $160.00 in order to make a profit. It spends $100.00 on supplies. The shop will charge $12.00 for every souvenir x it sells.

The inequality 12x + 100 > 160 expresses the number of souvenirs the shop must sell in order to make a profit.

Plot the points for which the shop will make a profit along the number line below.

Answer
x > 5

Answer Process
• The calculations are the same as in the previous queation.
• Because the inequality 12x + 100 > 160 specifies greater than, points plotted along the number line exclude 5, start with 6, and proceed higher from there.

Question
A souvenir shop must generate $163.00 in order to make a profit. It spends $100.00 on supplies. The shop will charge $12.00 for every souvenir x it sells.

The inequality 12x + 100 > 163 expresses the number of souvenirs the shop must sell in order to make a profit.

Plot the points for which the shop will make a profit along the number line below.

Answer
x > 5.25, starting with 6

Answer Process
See Equations.

12x + 100 > 163

12x + 100 – 100 = 163 – 100
• 100 is subtracted from each side of = sign
• subtraction is the opposite of addition

12x = 63
• 100 – 100 is the same as 0, so it disappears
• 163 – 100 is the same as 63

$\bf\displaystyle\frac{12x}{12}$ = $\bf\displaystyle\frac{63}{12}$
• each side of = sign is divided by 12
• division is the opposite of multiplication

x = 5.25
• $\bf\displaystyle\frac{12x}{12}$ is the same as 1x which is the same as x
• $\bf\displaystyle\frac{63}{12}$ is the same as $\bf\displaystyle\frac{21}{4}$ which is the same as 5.25 after toggle
• x has been isolated on the left side of = sign

x > 5.25, starting with 6
• Because there is no such thing as 5.25 souvenirs and the inequality 12x + 100 > 163 specifies greater than, points plotted along the number line exclude 5, start with 6, and proceed higher from there.

Input	Display	Comment
	blinker	clears screen

163 – 100	163-100	subtraction on right side of =

	63

$fraction$ 63 12	$\bf\displaystyle\frac{63}{12}$	division on right side of =

	$\bf\displaystyle\frac{21}{4}$

	5.25	Toggle

		Inequality

Question
A baseball card collector can spend up to $325.00 at a show. The show charges a $25.00 entrance fee. The collector plans to spend $75.00 on each card x at the show.

The inequality 75x + 25 ≤ 325 expresses the number of baseball cards the collector can buy. Plot the points representing all possible numbers of baseball cards the collector can buy along the number line below.

Answer
x ≤ 4

Answer Process
See Equations.

75x + 25 ≤ 325

75x + 25 – 25 = 325 – 25
• 25 is subtracted from each side of = sign
• subtraction is the opposite of addition

75x = 300
• 25 – 25 is the same as 0, so it disappears
• 325 – 25 is the same as 300

$\bf\displaystyle\frac{75x}{75}$ = $\bf\displaystyle\frac{300}{75}$
• each side of = sign is divided by 75
• division is the opposite of multiplication

x = 4
• $\bf\displaystyle\frac{75x}{75}$ is the same as 1x which is the same as x
• $\bf\displaystyle\frac{300}{75}$ is the same as 4
• x has been isolated on the left side of = sign

x ≤ 4
• Because the inequality 75x + 25 ≤ 325 specifies less than or equal to, points plotted along the number line include 4 and proceed lower from there. (Because the collector can choose to buy nothing, please note that point 0 should be plotted.)

Input	Display	Comment
	blinker	clears screen

325 – 25	325-25	subtraction on right side of =

	300

$fraction$ 300 75	$\bf\displaystyle\frac{300}{75}$	division on right side of =

	4

		Inequality

Question
A baseball card collector must spend less than $325.00 at a show. The show charges a $25.00 entrance fee. The collector plans to spend $75.00 on each card x at the show.

The inequality 75x + 25 < 325 expresses the number of baseball cards the collector can buy. Plot the points representing the number of baseball cards the collector can buy along the number line below.

Answer
x < 4

Answer Process
• The calculations are the same as in the previous question.
• Because the inequality 75x + 25 < 325 specifies less than, points plotted along the number line exclude 4, start with 3, and proceed lower from there. (Because the collector can choose to buy nothing, please note that point 0 should be plotted.)

Question
A baseball card collector must spend less than $325.00 at a show. The show charges an $80.00 entrance fee. The collector plans to spend $75.00 on each card x at the show.

The inequality 70x + 80 < 325 expresses the number of baseball cards the collector can buy.

Plot the points representing the number of baseball cards the collector can buy along the number line below.

Answer
x < 3.5, starting with 3

Answer Process
See Equations.

70x + 80 < 325

70x + 80 – 80 = 325 – 80
• 80 is subtracted from each side of = sign
• subtraction is the opposite of addition

70x = 245
• 80 – 80 is the same as 0
• 325 – 80 is the same as 245

$\bf\displaystyle\frac{70x}{70}$ = $\bf\displaystyle\frac{245}{70}$
• each side of = sign is divided by 70
• division is the opposite of multiplication

x = 3.5
• $\bf\displaystyle\frac{70x}{70}$ is the same as 1x which is the same as x
• $\bf\displaystyle\frac{245}{70}$ is the same as $\bf\displaystyle\frac{7}{2}$ which is the same as 3.5 after toggle
• x has been isolated on the left side of = sign

x < 3.5, starting with 3
• Because there is no such thing as 3.5 baseball cards and the inequality 70x + 80 < 325 specifies less than, points plotted along the number line exclude 4, start with 3, and proceed lower from there. (Because the collector can choose to buy nothing, please note that point 0 should be plotted.)

Input	Display	Comment
	blinker	clears screen

325 – 80	325-80	subtraction on right side of =

	245

$fraction$ 245 70	$\bf\displaystyle\frac{245}{70}$	division on right side of =

	$\bf\displaystyle\frac{7}{2}$

	3.5	Toggle

		Inequality

Practice – Questions
1. For x < 8, plot points for whole numbers along the number line below.

2. For x ≤ 8, plot points for whole numbers along the number line below.

3. For x > 8, plot points for whole numbers along the number line below.

4. For x ≥ 8, plot points for whole numbers along the number line below.

5. A kiosk must generate $200.00 in order to break even. It spends $151.00 on supplies. The kiosk will charge $7.00 for every item x it sells.

The inequality 7x + 151 ≥ 200 expresses the number of items the kiosk must sell in order to break even or make a profit.

Plot the points for which the kiosk will break even or make a profit along the number line below.

6. A kiosk must sell more than $200.00 in order to make a profit. It spends $151.00 on supplies. The kiosk will charge $7.00 for every item x it sells.

The inequality 7x + 151 > 200 expresses the number of items the kiosk must sell in order to make a profit.

Plot the points for which the kiosk will make a profit along the number line below.

7. A kiosk must sell more than $205.00 in order to make a profit. It spends $151.00 on supplies. The kiosk will charge $7.00 for every item x it sells.

The inequality 7x + 151 > 205 expresses the number of items the kiosk must sell in order to make a profit.

Plot the points for which the kiosk will make a profit along the number line below.

8. A tour group can spend up to $194.00 on a sightseeing trip. The tour bus has a one-time reservation fee of $50.00. The tour bus charges $36.00 for each tourist x.

The inequality 36x + 50 ≤ 194 expresses the number of tourists the tour group can afford.

Plot the points representing the number of tourists the tour group can afford along the number line below.

9. A tour group must spend less than $194.00 on a sightseeing trip. The tour bus has a one-time reservation fee of $50.00. The tour bus charges $36.00 for each tourist x.

The inequality 36x + 50 < 194 expresses the number of tourists the tour group can afford.

Plot the points representing the number of tourists the tour group can afford along the number line below.

10. A tour group must spend less than $184.00 on a sightseeing trip. The tour bus has a one-time reservation fee of $50.00. The tour bus charges $36.00 for each tourist x.

The inequality 36x + 50 < 184 expresses the number of tourists the tour group can afford.

Plot the points representing the number of tourists the tour group can afford along the number line below.

Practice – Answers
1. x < 8