Connexions

You are here: Home » Content » Maths Grade 10 Rought draft » Linear inequalities

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• FETMaths

This module is included inLens: Siyavula: Mathematics (Gr. 10-12)
By: Siyavula

Review Status: In Review

Click the "FETMaths" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Inside Collection (Textbook):

Textbook by: Nerk van Rossum. E-mail the author

Linear inequalities

Equations and inequalities: linear inequalities

Investigation : Inequalities on a Number Line

Represent the following on number lines:

1. x = 4 x = 4
2. x < 4 x < 4
3. x â‰¤ 4 x â‰¤ 4
4. x â‰¥ 4 x â‰¥ 4
5. x > 4 x > 4

A linear inequality is similar to a linear equation in that the largest exponent of a variable is 1. The following are examples of linear inequalities.

2 x + 2 â‰¤ 1 2 - x 3 x + 1 â‰¥ 2 4 3 x - 6 < 7 x + 2 2 x + 2 â‰¤ 1 2 - x 3 x + 1 â‰¥ 2 4 3 x - 6 < 7 x + 2
(1)

The methods used to solve linear inequalities are identical to those used to solve linear equations. The only difference occurs when there is a multiplication or a division that involves a minus sign. For example, we know that 8>68>6. If both sides of the inequality are divided by -2-2, -4-4 is not greater than -3-3. Therefore, the inequality must switch around, making -4<-3-4<-3.

Tip:

When you divide or multiply both sides of an inequality by any number with a minus sign, the direction of the inequality changes. For this reason you cannot divide or multiply by a variable.

For example, if x<1x<1, then -x>-1-x>-1.

In order to compare an inequality to a normal equation, we shall solve an equation first. Solve 2x+2=12x+2=1.

2 x + 2 = 1 2 x = 1 - 2 2 x = - 1 x = - 1 2 2 x + 2 = 1 2 x = 1 - 2 2 x = - 1 x = - 1 2
(2)

If we represent this answer on a number line, we get

Now let us solve the inequality 2x+2â‰¤12x+2â‰¤1.

2 x + 2 â‰¤ 1 2 x â‰¤ 1 - 2 2 x â‰¤ - 1 x â‰¤ - 1 2 2 x + 2 â‰¤ 1 2 x â‰¤ 1 - 2 2 x â‰¤ - 1 x â‰¤ - 1 2
(3)

If we represent this answer on a number line, we get

As you can see, for the equation, there is only a single value of xx for which the equation is true. However, for the inequality, there is a range of values for which the inequality is true. This is the main difference between an equation and an inequality.

Figure 3
Khan academy video on inequalities - 1

Figure 4
Khan academy video on inequalities - 2

Exercise 1: Linear Inequalities

Solve for rr: 6-r>26-r>2

Solution

1. Step 1. Move all constants to the RHS :
- r > 2 - 6 - r > - 4 - r > 2 - 6 - r > - 4
(4)
2. Step 2. Multiply both sides by -1 :

When you multiply by a minus sign, the direction of the inequality changes.

r < 4 r < 4
(5)
3. Step 3. Represent answer graphically :

Exercise 2: Linear Inequalities

Solve for qq: 4q+3<2(q+3)4q+3<2(q+3) and represent the solution on a number line.

Solution

1. Step 1. Expand all brackets :
4 q + 3 < 2 ( q + 3 ) 4 q + 3 < 2 q + 6 4 q + 3 < 2 ( q + 3 ) 4 q + 3 < 2 q + 6
(6)
2. Step 2. Move all constants to the RHS and all unknowns to the LHS :
4 q + 3 < 2 q + 6 4 q - 2 q < 6 - 3 2 q < 3 4 q + 3 < 2 q + 6 4 q - 2 q < 6 - 3 2 q < 3
(7)
3. Step 3. Solve inequality :
2 q < 3 Divide both sides by 2 q < 3 2 2 q < 3 Divide both sides by 2 q < 3 2
(8)
4. Step 4. Represent answer graphically :

Exercise 3: Compound Linear Inequalities

Solve for xx: 5â‰¤x+3<85â‰¤x+3<8 and represent solution on a number line.

Solution

1. Step 1. Subtract 3 from Left, middle and right of inequalities :
5 - 3 â‰¤ x + 3 - 3 < 8 - 3 2 â‰¤ x < 5 5 - 3 â‰¤ x + 3 - 3 < 8 - 3 2 â‰¤ x < 5
(9)
2. Step 2. Represent answer graphically :

Linear Inequalities

1. Solve for xx and represent the solution graphically:
1. 3x+4>5x+83x+4>5x+8
2. 3(x-1)-2â‰¤6x+43(x-1)-2â‰¤6x+4
3. x-73>2x-32x-73>2x-32
4. -4(x-1)<x+2-4(x-1)<x+2
5. 12x+13(x-1)â‰¥56x-1312x+13(x-1)â‰¥56x-13

2. Solve the following inequalities. Illustrate your answer on a number line if xx is a real number.
1. -2â‰¤x-1<3-2â‰¤x-1<3
2. -5<2x-3â‰¤7-5<2x-3â‰¤7
3. Solve for xx: 7(3x+2)-5(2x-3)>77(3x+2)-5(2x-3)>7

Illustrate this answer on a number line.


Content actions

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks