# SOLVING AND GRAPHING INEQUALITIES

If an inequality is in the form

(x < a)  or  (x > a)  or  (≤ 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.

## Solving Inequalities

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.

## Graphing Inequalities

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

2x - 4 + 4 ≤ 0 + 4

2x ≤ 4

Divide by 2 on both sides

2x/2 ≤ 4/2

x ≤ 2

So, 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

So, 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

4x - 12 + 12  ≥  0 + 12

4x  ≥  12

Divide by 4 on both sides

4x/4  12/4

x ≥ 3

So, 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

So, 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

2x - 3 + 3 <  1 + 3

2x < 4

Divide by 2 on both sides

2x/2 < 4/2

x < 2

So, 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

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

≤ -3

So, 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

-2x - 4 + 4 ≤ - 12 + 4

-2x ≤ - 8

Divide by -2 on both sides

-2x / (-2) ≤ - 8 / (-2)

≤ 4

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

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

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

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

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

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

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

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

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 