Solve and Check Inequalities

Introduction to solve and check inequalities:

Inequality is a one the concept of algebra.Inequality in the form of equation with greathearted or lesser than symbol. . Inequalities are described with the symbol of <, >, <=, and >=It is algebra expression. Inequalities is a combination of variables, numbers ,constants , conditions( < or >) with operations( + or -)

Four forms of checking inequalities:

1. Ax + By  <  C

2. Ax + By  >  C

3. Ax + By  <  C

4. Ax + By  >  C

Basic Concepts of Checking Inequalities:

Types of inequalities with examples

Less Than (<)

Example of solve and check inequalities:

x<6 br="br">
Here variable x is less than the value of six ,It mean x may contains the value of like 5,4,3,2,1,0,-1,-2,-3 … etc

Less Than (≤)

Example of solve and check inequalities:

x ≤ 8,

Here  variable x is less than or equal to 8 mean , that is x may contain the value of 8,7,6,5,4,3,2,……….. so on.

Greater than (>)

Example of solve and check inequalities:

x > 10,

Here the variable x  is great than value of  10, that is x may be any one of the following value 11,12,13……so on.

Greater Than or equal to (≥)

Example of solve and check inequalities:

x ≥ 3,

Here  variable x  is great than or equal to 3 means that, that is x may be contain any one of the following value 3,4,5,6……so on.

Example Problems in Inequalities:

Solving and check the inequalities

1. solve and check: 4x+5 < -2x+14

Solution:

4x+5 < -2x+14(given problem)

4x+5-5 < -2x+14-5( subtract both sid e by 5)

4x < -2x+9

4x+2x < -2x+2x+9( add both side by 2x)

6x < 9

6x /9< 9/9( divide both side by 9)

x < 3/2  in otherwords x<1 .5=".5" br="br">
So x values are may 1.4.1.3,1.2,1.1,1…..and so on

Check the ineuqlities:

Substitute x value to the given problem

Substitute x=1 in  4x+5 < -2x+14

4(1)+5 < -2(1)+14

5< -12

Condition are not satisfied here 5 is graeterthan the -12

So given problem is not inequalities.

Exampl 2:

Solve  and check x + 3 < 2x +7.

Sol:

x + 3 < 2x + 7

x +3 – 3 < 2x + 7 – 3 (Rule 1)

x < 2x + 4

x-2x < 2x-2x+4

- x<4 br="br">
x > – 1

The solution set is {0, 1, 2, 3…}

Verification:

Substitute x =0,1,2

Substitute x = 0 in given eqaution

X+3<2x br="br">
0+3<2 br="br">
3<7 br="br">
Here condition is satisfied so given equation is inequality

Solving Absolute Value Inequalities

Introduction to solving absolute value inequalities:

Solving absolute value inequalities is nothing but the value without considering its sign. Here we are going to learn how we are solving absolute value inequalities. The main difference between the equalities and inequalities is less than and greater than symbol. We will see some example problem for solving absolute value inequalities. For graphing the absolute value inequalities we will use the number lines.

Example Problems for Solving Absolute Value Inequalities:

Solving absolute value inequalities example 1:

Solve the following |x + 5| `gt=` 5

Solution:

Here we are having the inequality |x + 5| `gt= ` 5

If we write these inequalities in absolute value form we will get

x + 5 `gt=` 5 ………… (1) And - (x + 5) `gt=` 5 ………….. (2)

Equation 1:

x + 5 `gt=` 5

Add -5 on both sides we will get

x + 5 – 5 `gt=` 5 – 5

x `gt=` 0

Equation 2:

-(X + 5) `gt=` 5

Add+5 on both sides we will get

- x - 5 + 5 `gt=` 5 + 5

-x > = 10

Divide by -1 so

x `lt= ` -10

So the x lies between 0 and -10

-10 `gt=` x `gt=` 0

We can mark these inequalities using the number line.

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Solving Absolute Value Inequalities Example 2:

Solve the following |x + 3| `gt=` 9

Solution:

Here we are having the inequality |x + 3|` gt= ` 9

If we write these inequalities in absolute value form we will get

x + 3 `gt=` 9 ………… (1) And -(x + 3)` gt= ` 9 ………….. (2)

Equation 1:

x + 3 `gt= ` 9

Add -3 on both sides we will get

x + 3 – 3` gt=` 9 – 3

x` gt=` 6

Equation 2:

- (X + 3) `gt=` 9

Add +3 on both sides we will get

-x - 3 + 3 `gt=` 9 + 3

-x `gt= ` 12

Divide by -1 so

x `lt=` -12

So the x lies between 0 and -12

-12 `gt=` x` gt=` 0

These are the some examples for solving absolute value inequalities. We can mark these absolute value inequalities in number line.

Steps to Graphing Inequalities

Steps to graphing inequalities:

The graph displays the nature of relationship between the variables. One of the most useful graph that we obtain quite often is the graph. ax + by > c and ax + by < c are called  linear inequalities. When we combine the equation ax + by = c with ax + by > c we get                 ax + by `>=` c. Similarly by combining ax + by = c and ax + by < c we get ax + by `<=` c. Here also ax + by`<=` c and ax + by `>=` c are called inequalities.

Step to Draw the Graph of Inequalities:

The following steps are used to draw the graph of inequalities ax + by > c or ax  +  by < c.

Step 1:

Treat the in equation as if it were an equation. That is ax + by = c.

Step 2:

Draw the graph of ax + by = c.

Step 3:

The line ax + by = c divides the plane into two regions. Take any point in one of the regions (usually the origin (0, 0) is taken). Substitute this point in the given in. equalities. If the inequalities are satisfied then the region containing the point is the desired region and it is shaded. If the point does not satisfy the given in equation then the region not containing the point is the desired region and it is shaded.

The shaded region represents the given in equation. All the points in this region satisfy the inequalities and they are the solutions of the in equalities.

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Steps to Graphing Inequalities-example Problem:
Draw the graph of the in equation 2x + y `>=` 10.

Sol:

Draw the graph of 2x + y = 10.

2x + y=10 ? y =10 - 2x

X   1  5  0

y   8  0  10

Plot the points (1,8), (5,0) and (0,10) in the graph and join them to get the straight line 2x + y = 10. Take the point (0,0) and substitute in the in equation 2x + y `>=` 10.0 + 0 > 10 this is false. So the region not containing the point (0,0) is the desired region and it is shaded. The shaded region in the graph represents the in equation 2x + y `>=` 10. All the points in the shaded region satisfy 2x + y `>=` 10 and they are the solutions of 2x + y `>=` 10.