LINEAR INEQUALITIES IN ONE VARIABLE

Video Lesson

Solved Examples

Solve the each of following inequalities and graph the solution.

Example 1 :

2(x - 3) ≥ 3x – 4

Solution :

2(x - 3) ≥ 3x - 4

2x – 6 ≥ 3x – 4

Subtract 3x from both sides.

-x – 6 ≥ –4

Add 6 to both sides.

-x ≥ 2

Multiply both sides by -1.

x ≤ -2

The solution of the given inequality is (-∞, -2].

Example 2 :

Solution :

The least common multiple of the denominators (3, 6) is 6. Multiply both sides of the inequality by 6 to get rid of the denominators 3 and 6.

2x + 4 < 5x - 6

Subtract 5x from both sides.

-3x + 4 < -6

Subtract 4 from both sides.

-3x < -10

Divide both sides by -3.

x > ¹⁰⁄₃

The solution of the given inequality is (¹⁰⁄₃, +∞).

Example 3 :

2(x + 2) ≤ x – 5

Solution :

2(x + 2) ≤ x – 5

2x + 4 ≤ x – 5

Subtract x from both sides.

x + 4 ≤ –5

Subtract 4 from both sides.

x ≤ –9

The solution of the given inequality is (-∞, -9].

Example 4 :

Solution :

12 ≥ 2y - 5 ≥ -6

Add 5 to each term.

17 ≥ 2y ≥ -1

Divide each term by 2.

17 ≥ 2y ≥ -1

¹⁷⁄₂ ≥ y ≥ -¹⁄₂

-½ ≤ y ≤ ¹⁷⁄₂

The solution of the given inequality is [¹⁷⁄₂, -½].

Example 5 :

0 ≤ 2z + 5 < 8

Solution :

0 ≤ 2z + 5 < 8

Subtract 5 from each term.

-5 ≤ 2z < 3

Divide each term by 2.

-⁵⁄₂ ≤ z < ³⁄₂

The solution of the given inequality is [-⁵⁄₂, ³⁄₂).

Example 6 :

Solution :

The least common multiple of the denominators (4, 3) is 12. Multiply both sides of the inequality by 12 to get rid of the denominators 4 and 3.

3(x - 5) + 4(3 - 2x) < -24

3x - 15 + 12 - 8x < -24

-5x - 3 < -24

Add 3 to both sides.

-5x < -21

Divide both sides by -5.

x > ²¹⁄₅

The solution of the given inequality is (²¹⁄₅, +∞).

Example 7 :

Solution :

The least common multiple of the denominators (2, 5) is 10. Multiply both sides of the inequality by 10 to get rid of the denominators 2 and 5.

5(2y – 3) + 2(3y – 1) < 10y – 10

10y - 15 + 6y - 2 < 10y - 10

16y – 17 < 10y – 10

Subtract 10y from both sides.

6y - 17 < -10

Add 17 to both sides.

6y < 7

Divide both sides by 6.

y < ⁷⁄₆

The solution of the given inequality is (-∞, ⁷⁄₆).

Example 8 :

Solution :

Ther least common multiple of the denominators (6, 8) is 24. Multiply both sides of the inequality by 24 to get rid of the denominators 6 and 8.

4(3 - 4y) - 3(2y - 3) ≥ 48 - 24y

12 - 16y - 6y + 9 ≥ 48 - 24y

-22y + 21 ≥ 48 - 24y

Add 24y to both sides.

2y + 21 ≥ 48

Subtract 21 from both sides.

2y ≥ 27

Divide both sides by 2.

y ≥ ²⁷⁄₂

The solution of the given inequality is [²⁷⁄₂, +∞).

Example 9 :

Solution :

x - 4 - 4x ≤ 30 - 10x

-3x - 4 ≤ 30 - 10x

Add 10x to both sides.

7x - 4 ≤ 30

Add 4 to both sides.

7x ≤ 34

Divide both sides by 7.

x ≤ ³⁴⁄₇

The solution of the given inequality is (-∞, ³⁴⁄₇].

Example 10 :

Solution :

4 > 3y – 1 > -4

Add 1 to each term.

5 > 3y > -3

Divided each term by 3.

⁵⁄₃ > y > -1

-1 < y < ⁵⁄₃

The solution of the given inequality is (-1, ⁵⁄₃)

Example 11 :

-6 < 5t – 1 < 0

Solution :

-6 < 5t – 1 < 0

Add 1 to each term.

-5 < 5t < 1

Divide each term by 5.

-1 < t < ⅕

The solution of the given inequality is (-1, 1/5)

Example 12 :

Solution :

3(3 - x) + 2(5x - 2) < -6

9 - 3x + 10x - 4 < -6

7x + 5 < -6

Subtract 5 from both sides.

7x < -11

Divide botgh sides by 7.

x < -¹¹⁄₇

The solution of the given inequality is (-∞, -¹¹⁄₇).

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