SOLVING LINEAR INEQUALITIES IN ONE VARIABLE

Solving linear inequalities is also like solving linear equation in one variable.

Rules for Solving Inequalities

  • When we add, subtract, multiply or divide any non zero number on both sides of the inequality sign, we don't have to change the inequality sign.
  • When we multiply or divide some negative number on both side, we have to reverse the sign.

Problem 1 :

4+6x ≤ x+6x

Solution :

Simplify both sides of the inequality.

6x+4 ≤ 7x

Subtract 7x from both sides.

6x+4-7x ≤ 7x-7x

-x+4  0

Subtract 4 from both sides.

-x+4-4 ≤ 0-4

-x ≤ -4

Since we divide both sides by -1, change inequality > into <.

-x/-1  -4/-1

 4

Problem 2 :

m+16 > 8m+2

Solution :

Subtract 8m from both sides.

m+16-8m > 8m+2-8m

-7m+16 > 2

Subtract 16m from both sides.

-7m+16-16 > 2-16

-7m > -14

Divide both sides by -7.

-7m/-7 > -14/-7

m < 2

Problem 3 :

2r-5 > 2r-5

Solution :

Subtract 2r from both sides.

2r-5-2r > 2r-5-2r

-5 > -5

Since, -5 and -5 are equal the condition will not exists for any value of r. So, there is no solution for x.

Problem 4 :

5x-1 ≥ 13-2x

Solution :

Add 2x to both sides.

5x-1+2x ≥ 13-2x+2x

7x-1 ≥ 13

Subtract 13 to both sides.

7x-1-13 ≥ 13-13

7x-14 ≥ 0

7x ≥ 14

x ≥ 2

Problem 5 :

6-4n < -1-4n

Solution :

Simplify both sides of the inequality.

-4n+6 < -4n-1

Add 4n to both sides.

-4n+6+4n < -4n-1+4n

6 < -1

Since this is a false condition, there is no solution for n.

Problem 6 :

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

Solution :

Simplify both sides of the inequality.

-4n > -7n-9

Add 7n to both sides.

-4n+7n > -7n-9+7n

3n > -9

Divide both side by 3

3n/3 > -9/3

n > -3

Problem 7 :

1+2m ≥ 8+m

Solution :

Subtract –m from both sides

1+2m-m ≥ 8+m-m

1+m ≥ 8

m ≥ 8-1

m ≥ 7

Problem 8 :

p-1 > 13+3p

Solution :

Subtract 3p from both sides.

p-1-3p > 13+3p-3p

-2p-1 > 13

-2p > 13+1

-2p > 14

Since we divide by -2, change inequality > as <.

-2p/-2 > -14/2

p < -7

Problem 9 :

7b+3b > -8+6b

Solution :

Subtract -6b from both sides.

7b+3b-6b > -8+6b-6b

4b > -8

Divide by 4 on both sides.

b > -2

Problem 10 :

2b-7 ≥ -14+2b

Solution :

Subtract -2b from both sides.

2b-7-2b ≥ -14+2b-2b

-7 ≥ -14

So, there are all real numbers for b.

Problem 11 :

-3k-2 ≥ -k+10

Solution :

Add k from both sides.

-3k-2+k ≥ -k+10+k

-2k-2 ≥ 10

-2k ≥ 10+2

-2k ≥ 12

Since we divide by -2, change inequality  into ≤ .

-2k/-2 ≥ 12/-2

≤ -6

Problem 12 :

-8x-3 > -3-8x

Solution :

Add 8x from both sides.

-8x-3+8x > -3-8x+8x

-3 > -3

Since, -3 and -3 are equal the condition will not exists for any value of x. So, there is no solution for x.

Problem 13 :

7+6x ≥ 7x+6x

Solution :

Simplify both sides of the inequality.

7+6x ≥ 13x

Subtract 13x to both sides.

7+6x-13x ≥ 13x-13x

7-7x ≥ 0

Subtract -7 to both sides.

7-7x-7 ≥ 0-7

-7x ≥ -7

Since we divide by -7, change inequality  into ≤ .

-7x/-7 ≥ -7/-7

x ≤ 1

Problem 14 :

-15+8x > 8x-3x

Solution :

Simplify both sides of the inequality.

-15+8x > 5x

Subtract 5x from both sides.

-15+8x-5x > 5x-5x

-15+3x > 0

3x > 15

x > 5 

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