**Solving Linear Inequalities in One Variable Examples :**

**In this section, we will learn, how to solve linear inequalities in one variable. **

Let a be a non zero real numbers and x be a variable.

Then, the inequality of the form

ax + b < 0

ax + b ≤ 0

ax + b > 0

ax + b ≥ 0

are known as linear inequalities in one variable.

**Rule 1 : **

Same number may be added to (or subtracted from) both sides of an inequality without changing the sign of inequality.

**Rule 2 : **

Both sides of an inequality can be multiplied (or divided) both by the same positive real number without changing the sign of inequality. However, the sign of inequality is revered when both sides of an inequality are multiplied or divided by the negative number.

**Rule 3 :**

Any term of an inequality may be taken to the other side with its sign changed without affecting sings of inequality.

Let us see some examples based on the above concept.

**Example 1 :**

Solve 5x - 3 < 3x + 1 when

(i) when x is a real number

(ii) when x is an integer

(iii) when x is a natural number

**Solution :**

**(i) When x is a real number :**

5x - 3 < 3x + 1

Subtract 3x from each side.

2x - 3 < 1

Add 3 to each side.

2x < 4

Divide each side by 2

x < 2

Because x is real number, the solution set is

(-∞, 2)

**(ii) When x is an integer :**

We have already solved for x in the given inequality.

That is

x < 2

Because x is an integer, the solution set is

{...............,-4, -3, -2, - 1, 0, 1, 2, 3,...............}

**(iii) When x is a natural number : **

x < 2

Because x is a integer, the solution set is

{ 1 }

**Example 2 :**

Solve for x :

3x + 17 ≤ 2(1 - x)

**Solution :**

3x + 17 ≤ 2(1 - x)

3x + 17 ≤ 2 - 2x

Add 2x to each side.

5x + 17 ≤ 2

Subtract 17 from each side.

5x ≤ - 15

Divide each side by 5.

x ≤ - 3

So, the solution set is

(-∞, -3]

**Example 3 :**

Solve for x :

2(2x + 3) - 10 ≤ 6(x - 2)

**Solution :**

2(2x + 3) - 10 ≤ 6(x - 2)

4x + 6 - 10 ≤ 6x - 12

4x - 4 ≤ 6x - 12

Subtract 6x from each side.

-2x - 4 ≤ - 12

Add 4 to each side.

-2x ≤ - 8

Divide each side by (-2).

x ≥ 4

So, the solution set is

[4, ∞)

**Example 4 :**

Solve for x :

3x - 7 > x + 1

**Solution :**

3x - 7 > x + 1

Subtract x from each side.

2x - 7 > 1

Add 7 to each side.

2x > 8

Divide each side by 2.

x > 4

So, the solution set is

(4, ∞)

**Example 5 :**

Solve for x :

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

**Solution :**

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

-x + 3 + 4 < 5 - 2x

-x + 7 < 5 -2x

Add 2x to each side.

x + 7 < 5

Subtract 7 from each side.

x < - 2

So, the solution set is (-∞, -2).

After having gone through the stuff given above, we hope that the students would have understood, how to solve linear inequalities in variable.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**