**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.**

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

Question

*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

=

• each side of = sign is divided by 12

• division is the opposite of multiplication

x = 5

• is the same as 1x which is the same as x

• 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.

Calculator Click |
What You See |
Comment |

blinker | clears screen | |

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

60 | ||

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.

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

Question

*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

=

• each side of = sign is divided by 12

• division is the opposite of multiplication

x = 5.25

• is the same as 1x which is the same as x

• is the same as 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.

Calculator Click |
What You See |
Comment |

blinker | clears screen | |

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

63 | ||

63 12 | division on right side of = | |

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

=

• each side of = sign is divided by 75

• division is the opposite of multiplication

x = 4

• is the same as 1x which is the same as x

• 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.)

Calculator Click |
What You See |
Comment |

blinker | clears screen | |

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

300 | ||

300 75 | division on right side of = | |

4 | ||

## Inequality |

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

Question

*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.)

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

Question

*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

=

• each side of = sign is divided by 70

• division is the opposite of multiplication

x = 3.5

• is the same as 1x which is the same as x

• is the same as 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.)

Calculator Click |
What You See |
Comment |

blinker | clears screen | |

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

245 | ||

245 70 | division on right side of = | |

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

2. x ≤ 8

3. x > 8

5. x ≥ 7

6. x > 7, starting with 8

7. x > 7.71, starting with 8

8. x ≤ 4

9. x < 4, starting with 3

10. x < 3.72, starting with 3