Solving linear inequalities is also like solving linear equation in one variable.
Rules for Solving Inequalities
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
x ≥ 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
k ≤ -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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