**Solving and Graphing Inequalities :**

In this section, we will learn, how to solve and graph linear inequalities.

If an inequality is in the form

(x < a) or (x > a) or (x ≤ a) or (x ≥ a),

where a is a constant, we can easily sketch the graph of the inequality.

In case, the inequality is in the form

ax + b < bx + d,

where a, b and d are constants, we have to solve for x and sketch the graph.

**Step 1 :**

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

**Step 2 :**

Both sides of an inequality can be multiplied or divided by 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 a negative number.

**Step 3 :**

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

In graphing inequalities in one variable on a number line, we have to follow the steps given below.

**Step 1 : **

If we have one of the signs like < (less than) or > (greater than), we have to use the unfilled circle.

**Step 2 : **

If we have one of the signs like ≤ (less than or equal to) or ≥ (greater than or equal to), we have to use the filled circle.

**Example 1 :**

Solve the following linear inequality and graph.

2x - 4 ≤ 0

**Solution :**

**2x - 4 ≤ 0**

**Add 4 on both sides**

**2x - 4 + 4 ****≤ 0 + 4**

**2x ****≤ 4**

**Divide by 2 on both sides**

**2x/2 ****≤ 4/2**

**x ≤ 2**

**Hence, any real number less than or equal to 2 is a solution of the given equation.**

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

**Example 2 :**

Solve the following linear inequality and graph.

-3x + 12 < 0

**Solution :**

-3x + 12 < 0

**Subtract 12 on both sides**

-3x + 12 - 12 < 0 - 12

-3x < -12

Divide by -4 on both sides

-3x/(-3) < -12/(-3)

x < 4

**Hence, any real number less 4 is a solution of the given equation.**

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

**Example 3 :**

Solve the following linear inequality and graph.

4x - 12 ≥ 0

**Solution :**

4x - 12 ≥ 0

**Add 12 on both sides**

4x - 12 + 12 ≥ 0 + 12

4x ≥ 12

Divide by 4 on both sides

4x/4 ≥ 12/4

x ≥ 3

**Hence, any real number greater than or equal to 3 is a solution of the given equation.**

The solution set of the given inequality is [3, ∞).

**Example 4 :**

Solve the following linear inequality and graph.

7x + 9 > 30

**Solution :**

7x + 9 > 30

**Subtract 9 on both sides**

7x + 9 - 9 > 30 - 9

7x > 21

Divide by 7 on both sides

7x/7 > 21/7

x > 3

**Hence, any real number greater than 3 is a solution of the given equation.**

The solution set of the given inequality is (3, ∞).

**Example 5 :**

Solve the following linear inequality and graph.

5x - 3 < 3x + 1

**Solution :**

5x - 3 < 3x + 1

**Subtract 3x on both sides**

**5x - 3 - 3x < 3x + 1 - 3x **

**2x - 3 < 1 **

**Add 3 on both sides**

**2x - 3 + 3 < 1 + 3**

**2x < 4**

**Divide by 2 on both sides**

**2x/2 < 4/2**

**x < 2**

**Hence, any real number lesser than 2 is a solution of the given equation.**

The solution set of the given inequality is (2, ∞).

**Example 6 :**

Solve the following linear inequality and graph.

3x + 17 ≤ 2(1 - x)

**Solution :**

3x + 17 ≤ 2(1 - x)

3x + 17 ≤ 2 - 2x

**Add 2x on both sides**

**3x + 2x + 17 ** ≤ 2 - 2x + 2x

5x + 17 ≤ 2

Subtract 17 on both sides

5x + 17 - 17 ≤ 2 - 17

5x ≤ -15

Divide by 5 on both sides

5x/5 ≤ -15/5

x ≤ -3

**Hence, any real number lesser than or equal to -3 is a solution of the given equation.**

The solution set of the given inequality is (-∞ , -3].

**Example 7 :**

Solve the following linear inequality and graph.

2(2x + 3) - 10 ≤ 6 (x - 2)

**Solution :**

2(2x + 3) - 10 ≤ 6 (x - 2)

4x + 6 - 10 ≤ 6 x - 12

4x - 4 ≤ 6 x - 12

Subtract 6x on both sides

4x - 4 - 6x ≤ 6 x - 12 - 6x

-2x - 4 ≤ - 12

Add 4 on both sides

-2x - 4 + 4 ≤ - 12 + 4

-2x ≤ - 8

Divide by -2 on both sides

-2x / (-2) ≤ - 8 / (-2)

x ≤ 4

**Hence, any real number lesser than or equal to 4 is a solution of the given equation.**

The solution set of the given inequality is (-∞ , 4].

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

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

HTML Comment Box 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**

**Sum of all three four digit numbers formed using 0, 1, 2, 3**

**Sum of all three four digit numbers formed using 1, 2, 5, 6**