SOLVING EQUATIONS AND INEQUALITIES WITH VARIABLES ON BOTH SIDES

Solve the following :

Problem 1 :

6x - 5x = 0

Solution :

6x - 5x = 0

x = 0

Problem 2 :

17 = n + 2 + 4n

Solution :

17 = n + 2 + 4n

17 = 5n + 2

Subtracting 2 on both sides

17 - 2 = 5n

15 = 5n

Dividing by 5 on both sides.

15/5 = n

n = 3

x = 0

Problem 3 :

-2(2m + 7) = -28 - 6m

Solution :

-2(2m + 7) = -28 - 6m

Using distributive property

-4m - 14 = -28 - 6m

Adding 6n on both sides, we get

-4m + 6m = -28 + 14

2m = -14

Dividing by 2 on both sides

m = -14/2

m = -7

Problem 4 :

25 + r = -5(2 + 8r) + 6r

Solution :

25 + r = -5(2 + 8r) + 6r

Using distributive property

25 + r = -10 - 40r + 6r

25 + r = -10 - 34r

25 + 10 = -34r - r

35 = -35r

r = -35/35

r = -1

Problem 5 :

-6n - 12(-11n + 8) = -10(n - 4)

Solution :

-6n - 12(-11n + 8) = -10(n - 4)

Using distributive property

-6n + 132n - 96 = -10n + 40

Combining like terms,

126n - 96 = -10n + 40

126n + 10n = 40 + 96

136n = 136

n = 136/136

n = 1

Problem 6 :

-28x + 6 = -3(1 + 7x) - 7x

Solution :

-28x + 6 = -3(1 + 7x) - 7x

Using distributive property

-28x + 6 = -3 - 21x - 7x

-28x + 6 = -3 - 28x

-28x + 28x = -3 - 6

0x = -9

There is no solution for x.

Problem 7 :

4 + 6x  ≤ x + 6x

Solution :

4 + 6x  ≤ x + 6x

4 + 6x ≤ 7x

Subtracting 6x on both sides

4 ≤ 7x - 6x

4 ≤ x

Problem 8 :

m + 16 > 8m + 2

Solution :

m + 16 > 8m + 2

16 > 8m - m + 2

16 > 7m + 2

16 - 2 > 7m

14 > 7m

Dividing by 7 on both sides.

14 / 7 > m

2 > m

Problem 9 :

2r - 5 > 2r - 5

Solution :

2r - 5 > 2r - 5

Applying infinite values of r will satisfy the inequality, so the given inequality will have infinite number of solutions.

Problem 10 :

5x - 1 ≥ 13 - 2x

Solution :

5x - 1 ≥ 13 - 2x

Adding 2x on both sides and adding 1 on both sides, we get

5x + 2x ≥ 13 + 1

7x ≥ 14

Dividing by 7 on both sides

x ≥ 14/7

x ≥ 2

Problem 11 :

-7n + 3n > -9 - 7n

Solution :

-7n + 3n > -9 - 7n

Combining like terms, we get

-4n > -9 - 7n

-4n + 7n > -9

3n > -9

n > -9/3

n > -3

Problem 12 :

-15 + 8x > 8x - 3x

Solution :

-15 + 8x > 8x - 3x

-15 + 8x > 5x

Subtracting 8x on both sides

-15 > 5x - 8x

-15 > -3x

Dividing by -3 on both sides

-15/(-3) > -3x/(-3)

5 > x

Problem 13 :

The width of a rectangle is x cm. The length is 1·5 cm more than the width. The perimeter of the rectangle is 17 cm. Write down an equation satisfied by x and solve it to find x

Solution :

Let x be the width of the rectangle. Length will be x + 1.5

Perimeter of the rectangle = 17 cm

2(length + width) = 17

2(x + 1 .5 + x) = 17

(2x + 1.5) = 17/2

2x + 1.5 = 8.5

2x = 8.5 - 1.5

2x = 7

x = 7/2

x = 3.5

So, the value of x, that is width is 3.5 cm and length is 5 cm.

Problem 14 :

The diagram shows a garden in the shape of a rectangle.

All measurements are in meters. The perimeter of the garden is 32 meters. Work out the value of x

Solution :

Perimeter of the garden = 32 m

length = 4 + 3x and width = x + 6

2(4 + 3x + x + 6) = 32

4x + 10 = 32/2

4x + 10 = 16

4x = 16 - 10

4x = 6

x = 6/4

= 1.5

So, the value of x is 1.5