3(x - 2) - $\frac{\mathrm{x}}{3}$ + 6 > 0

Get rid of the fraction by multiplying both sides of the inequality by the denominator (3).

3 x 3(x - 2) - 3 x $\frac{\mathrm{x}}{3}$ + 3 x 6 > 3 x 0
9(x - 2) -x + 18 > 3 x 0
9 x x -2 x 9 -x > 0 - 18
9x - 18 -x >-18
9x -x > -18 + 18
8x > 0

Divide both sides by 8 to make x stand alone.

x > $\frac{0}{8}$
x > 0

Note: 0 divided by any number is 0.

You can substite x = 0 into the equation to confirm you get 0.

3(0 - 2) - $\frac{0}{3}$ + 6 > 0

3(- 2) - 0 + 6 > 0
3 x -2 + 6 > 0
-6 + 6 > 0
0 > 0

Expand the bracket

- (7x - 3) = -1 x (7x - 3)

Apply a(b - c) = a x b + a x - c

-1 x (7x - 3) = -1 x 7x -1 x - 3

- x - = + hence -1 x - 3 = + 3

-1 x (7x - 3) = -1 x 7x -1 x - 3 = - 7x + 3

5x - (7x - 3) ≤ 9

5x - 7x + 3 ≤ 9

Simplify

-2x ≤ 9 - 3

-2x ≤ 6

Divide both sides by - 2

NOTE: When you divide by negative, the signs changes to the opposite direction

Cancel out

x ≥ -3

Solve: (1-x)÷3<4

Get rid of the denominators by multiplying both sides of the equation by L.C.M. Every number is divisible by 1.

The L.C.M of 3 and 1 is 3

Divide both sides by -1.

NOTE: When dividing x by negative, the sign must change x > -11

Alternatively you can take the x to the right hand side and the numbers to the left.

Which reads -11 is less than x

Hence if -11 is less than x means x is also greater than – 11.

Hence x > -11

Let’s test

Hence x > -11 is correct

Testing x < -11 is option too. Numbers less than -11 are -12, -13, -14, -15 etc.

when x = -12

Plotting inequalities on number lines

Inequalities can be represented on a number line.

Use a hollow dot for:
< and >

Use a solid dot for:
≤ and ≥

Example

This shows x is greater than or equal to : -1

x ≥ -1

Example

This shows x is less than : 2

x < 2

Example

This shows that x is greater than or equal to -1 and x is less than 3 :

x ≥ -1 and x < 3

Which can be written as -1 ≤ x < 3, which reads -1 is less than or equal to x and x is less than 3

Notice that we can change the order of the first part and it still means the same thing.

x ≥ -1 means as -1 ≤ x

3 - (3x+4) ≤ -4

Concept

a(b+c) = axb+axc

-a(b+c) = -a x b + -axc

-(a+b) = -1(a+b) = -1xa + -1xb

⇒ - (3x+4) = -1(3x+4) = -1 x 3x + -1x4 = -3x-4

3 - (3x+4) ≤ -4

⇒ 3 - 3x-4 ≤ -4

⇒ 3 -4 - 3x ≤ -4

⇒ -1 - 3x ≤ -4

Moving the - 3x to the right side of the equation and the -4 to the left

⇒ -1+4 ≤ 3x

3 ≤ 3x

Divide both sides by 3

3/3 ≤ 3x/3

⇒ 1 ≤ x

You can reverse the order and reverse the inequality sign

⇒ x ≥ 1