# SOLVE LINEAR INEQUALITIES

Solve linear inequalities :

A statement involving variable or variables and the sign of inequality like <, >, ≤ or ≥ is called an inequality.

Here we are going to see how to solve linear inequalities.

• Solving linear inequalities in one variable
• Solving linear inequalities in two variables

Linear equations 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 and ax + b  0 are known as linear equations in one variable.

Rules followed in solving linear equations 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 the following linear inequality

2x - 4  ≤  0

Solution :

2x - 4  ≤  0

2x - 4 + 4  ≤   0 + 4

2x  ≤  4

Divide by 2 on both sides

2x/2 ≤  4/2

≤  2

By graphing this in the number line, we can split this as two intervals. (-∞, 2] and [2, ∞)

Now we need to select one of the values from the above intervals and apply those values instead of x in the given question 2x - 4  ≤  0.

The values from which interval makes the given inequality true is the solution set.

 (-∞, 2]Let x = 02(0) - 4  ≤  0- 4 ≤  0 True [2, ∞)Let x = 32(3) - 4  ≤  06 - 4 ≤  02 ≤  0 False

Hence (-∞, 2] is the solution set.

Linear equations in two variables :

If a, b, c are real numbers, then the equation ax + by + x = 0 is called a linear equation in two variables.Whereas the inequalities ax + by < c, ax + by ≤ c, ax + by > c and ax + by  c are known as linear equations in two variables.

Steps followed in solving linear equations in two variables :

Step 1 :

Convert the given inequality, say ax + by ≤ c, into the equation ax + by = c which represents a straight line in xy plane.

Step 2 :

Put y = 0 in order to find the x-intercept and similarly put x = 0 to find y-intercept. Now draw the straight by using these intercepts.

Step 3 :

Choose a point and substitute its coordinates in the inequality, if the inequality is satisfied, then shade the portion of the plane contains the chosen point, otherwise shade the portion which does not contain the desired solution set.

Example 1 :

Solve the following inequality graphically

2x - y    1

Solution :

Consider the linear inequality as linear equation

2x - y = 1

 x-interceptput y = 02x - 0 = 1 2x = 1x = 1/2 = 0.5(0.5, 0) y-interceptput x = 02(0) - y = 1 -y = 1y = -1(0, -1)

By using the above two points we can draw a straight line.Now we need to select a point say (2, -2)

2x - y    1

2(2) - (-2)    1

4 + 2   1

1

Since the chosen point satisfies the given inequality, we can shade that portion.

After having gone through the stuff given above, we hope that the students would have understood "Solve linear inequalities".

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

WORD PROBLEMS

HCF and LCM  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