How to solve a inequality

An inequality is a statement about the relative size or order of two objects, or about whether they are the same or not (See also: equality).these points covered when linear equations explained

• The notation a < b means that a is less than b.
• The notation a > b means that a is greater than b.
• The notation a ≠ b means that a is not equal to b, but does not say that one is greater than the other or even that they can be compared in size.

In each statement above, a is not equal to b. These relations are known as strict inequalities. Let's see an example on this from 9th grade math

Question:-

Solve the inequality

13 ≤ 3y+4 < style="font-weight: bold; color: rgb(51, 51, 255);">

Answer:-


We have to find the range of y.

So we isolate y.

13-4 ≤ 3y+4-4 < 18-4

9 ≤ 3y < 14

Noe divide each term by 3.

we get 3 ≤ y < 14/3 is the answer

similarly we can get more free answers to math problems

Problem on Distance Formula

The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. The hypotenuse of the triangle will be the distance between the two points.

Lets see an example on this from college algebra tutorial


Find the distance between(-3,-2)and (1,7)

here we have

x1=-3 ,y1=-2 and x2=1 and y2=7


We know the distance formula

D= √(x₂-x₁)²+(y₂-y₁)²

x₂ and y₂ are the x,y coordinates for one point
x₁and y₁ are the x,y coordinates for the second point
d is the distance between the two points
one of the Discrete math Tutor is solving the above problem here

D =√(-3-1)²+(-2-7)²

= √ (-4)² +(-9)²

= √ 16 + 81

= 9.848

So the distance is 9.85 approximately

Question on Calculating Simple Interest

In simple term the word Simple Interest refers just the amount of money paid on a loan. It's the easiest type of interest or most basic type of interest. In order to understand how various types of transactions work, it helps to have a complete understand the terms used.

For example: you may pay interest on a car loan, and it is very much necessary and important to understand how interest works. Better yet, your bank may be paying you interest on your deposits – and you can maximize your earnings by knowing more about interest.

Here S.I refers to simple interest.
Formula for S.I =

P x T x R
  100

Question: If 11,000,00 dollars is invested. Part of the 11,000,00 is invested 4% and the rest at 7%.
If the investment earns $888.00 per year,how much was invested at each rate?

Answer :

Principal time rate of interest simple interest earned
   x      1            4%              0.04x
1000-x    1            7%             .07(11000-x)

Simple interest = PTR
Where P is the Principal. T is the time and R is the rate of interest
SI = 680
Therefore

0.04x + 0.07(11000-x) = 680
0.04x + 770 - 0.07x   = 680

     -0.03x + 770   = 680
     -0.03x         = 680 - 770
     ------           ----------  
     -0.03            -0.03
          x         = 90/0.03
          x         = 3,000

x = 3,000
so, 3,000 is invested at 4% and 8,000 is the principal invested at 7%

A problem on Logarithms

Topic:-Logarithms

The logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number.
For example, the logarithm of 1000 to the base 10 is 3, because 3 is how many 10s you must multiply to get 1000: thus 10 × 10 × 10 = 1000; the base 2 logarithm of 32 is 5 because 5 is how many 2s one must multiply to get 32: thus 2 × 2 × 2 × 2 × 2 = 32. In the language of exponents: 10³ = 1000, so log₁₀1000 = 3, and 2⁵ = 32, so log₂32 = 5.
Let's see a problem on logarithms.

Question:-

solve 6^x+5=3^x-4

Answer:-

Given 6x+5=3x-4

log 6 x+5=log 3x-4

(x+5)log6 = (x-4)log3

xlog6+5log6 = xlog3-4log3

xlog6-xlog3= -4 log3-5log6

x[log6-log3]= -[4log3+5log6]

xlog 6/2 = -[log 34+log65]

xlog3= -log34*65

     -log(34*65)
x=  -----------------
       log3

x=-log3(34*65)

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problem on logarithms

Topic:-logarithms

To solve logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number.
Example logarithm problems : 1000 to the base 10 is 3, because 3 is how many 10s you must multiply to get 1000: thus 10 × 10 × 10 = 1000; the base 2 logarithm of 32 is 5 because 5 is how many 2s one must multiply to get 32: thus 2 × 2 × 2 × 2 × 2 = 32. In the language of exponents: 10³ = 1000, so log₁₀1000 = 3, and 2⁵ = 32, so log₂32 = 5.
The logarithm of x to the base b is written log_b(x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y,

An important feature of logarithms is that they reduce multiplication to addition, by the formula:

This math help is having a example

Question:-

         y        
solve log2---- = 4          
         3

Answer:-

         y        
here  log2---- = 4          
         3

log rule:-

logba = x

bx=a

So,We get
             y
   24= ---
             3

            y
  2*2*2*2= ---
            3
           y
     16 = ---
           3

    16*3=y

    So y=48 is the Answer.

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Calculating Area by Integration

Integration is an important concept in calculus where for a given relations and functions, x is a real variable and for the interval, integral will be calculated. Here is one such example.

Topic : Finding an area of the given function by Integration
Definite and indefinite integrals are the two special integrals.

Problem : Given function y = x² - 5x - 6. Find the area by Integration when function is enclosed by x - axis at the interval (0,4)

Solution :
The area is given by the formula

area = _a∫^b f(x) dx = _a∫^b y dx

given y = x² - 5x - 6
enclosed by x - axis, x = 0 and x = 4
Therefore,

Area = _{x = 4}∫^{x = 0} y dx

= ₀∫⁴ (x² - 5x - 6) dx

= [x³/3 - 5x²/2 - 6x _x=0]^x=4

= 4³/3 - 5(4)²/2 - 6(4)

= 64/3 - 80/2 - 24

= (128 - 240 - 144)/6

= (128 - 384)/6

= -256/6


So area is approximately 42.5 squnits

If you have any queries related to the above problem or the concept please feel free to contact calculus help.

Question on Finding Volume of the Solid, Using Shell Method for setup and Evaluating the integral

Here is a Post on Finding Volume of a Solid by shell Method set up and Evaluating the integral.

Topic : Volume of a Solid

On Integration also one can find out volume of any solid.

Question : Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis
y = 2x, y = 4, x = 0

Solution :

We know that the volume of the solid formed by rotating the area between the curves of f(x)
and g(x) and the lines x = a and x = b about the y-axis is given by







The graph of the line “ y = 2x” and “y = 4” between the two axes is denoted by the shaded region in the graph drawn below:






















So, the volume of the shaded region when revolved around y-axis is given by:





















Hence, the volume of the solid generated by revolving the curves “ y = 2x” and “y = 4” around y axis in the interval [0,2] is “16π /3”.


If you have any queries do write to our calculus help.