inequalities

Inequality is a collection of operators which is used to represent the inequality of algebraic equations. An inequality is a statement of algebraic expression to calculate the value of unknown variables. In general aspect we can say that inequality is used to calculate the algebraic expression that is not same in both sides of equal sign. The term inequality can be applied to any type of statement by using the various types of symbol like ‘>’ (greater then), ‘<’ (less then), ‘<=’ (less then equal to), ‘>=’ (greater then equal to) and so on. The concept of inequality helps the students of Grade VII to understand basic concepts of mathematics.
Here we show you the fundamental properties of inequalities to understand the concept of inequalities:
a)      x, y and z are the real numbers if x ≤ y then x + z ≤ y + z.
b)      x, y and z are the positive real numbers if x ≤ y then xz ≤ yz.
A solution of an inequality is a number which is substituted for the variable makes the inequality a true statement. In the mathematics there are various properties defined for inequality to solve equations. In the next session we are going to discuss Multistep problems.
a) Transitive property: if a > b and b > c then a > c.
b) Addition property: if a > b then a + c > b + c.
c) Multiplication property: if a > b then ab > ac.
d) Subtraction property: if a > b then a – c > b – c.
The above given properties of inequality helps the students to Graphing inequalities into the graph. Inequalities can be performed by solving the inequalities into the algebraic expressions. There are some rules given below:
a)      Adding and subtracting the same number on both sides.
b)      After performing the above rule interchange the sides and changing the orientation of the given inequality symbols.
c)      If needed, then perform the multiplication and division of same positive or negative number on both sides of equal sign then changing the orientation of the inequality symbol.

In the next session we are going to discuss Multistep problems. 

equations

This unit is for the students of Grade VII. In this unit we will learn about equations. We must remember that equations with variables, are inter related terms. We form the equations to express the mathematical statements in form of the expressions. The mathematical expressions joined with the numerical operators are called equations, when there exist two sides of the equation LHS and the RHS. Also by word equation we mean whatever placed on the left side of the equation is equal to the expression placed on the right side of the equation. By word variable, we mean the unknown value which may change every time. When we solve an equation with variable ‘x’ and another equation with same variable, then value of ‘x’ may change from equation to equation, which may satisfy the equation.
The basic purpose of solving any equation is to find the value of the value of the variable which is unknown in the given equation. When we say that we need to find the value of the unknown variable in the equation, we mean that the value we calculate must satisfy the equation. Thus we say that the  when we put the value of the unknown variable in the equation,  the value we get after solving both the sides of the equation must be equal.  (know more about cbse board books, here)
There are different ways to solve the equations and finding the values for the unknown variables in the equation. To find the value, we may adopt hit and trial method, where we put the values of the variables and then check it that particular value satisfies the given equation or not. The value that satisfies the given equation actually is the required solution to the given equation.
Another method to solve the equation is by shifting the variable to one side and all the constants to another side of the equation and  In the next session we will discuss about  Arithmetic sequences. 

math blog on grade VII

Dear kids,Previously we have discussed about how to simplify rational numbers and in this session we will talk about inequalities and problem related to inequalities and graphing inequalities problems, of grade VII of gujarat state education board.  We will learn here how to graph any inequality on the number line and how to solve them, You can take inequality solver help if needed.

An inequality is a type of linear equation in which there are two different expressions on both sides of a particular symbol either equality or the symbol of inequality. This symbol shows the relationship between these two expressions that how they are related to each other whether they are equal or have some comparative relation in them. In inequality, there are some of the symbols which are used in inequalities to show the relationship between the expressions. Say in equation 2x = 3y, here is the relation of equality between two expressions 2x and 3y. In a similar way the inequality 2 x > 3 y, shows that the value of left side expression is larger than that of right side of expression.

If an expression is greater than any other expression then in the notation, it will come in the right and smaller expression will come in the left of the inequality symbol. The inequality is also same as the number line notation. The symbol less than (<) is used to represent comparatively less value and (>) is used for larger value.

We can say here, that 4 is greater than -1, because 4 is on the right side of -1 (or -1 is on the left of 4). We write it as 4 > - 1 or as − 1 < 4. Let say for basic purpose two different expressions are as x and y, then:
                y > x       left side expression is greater than that of right side of expression.
                y < x       left side expression is less than that of right side of expression.
                y = x       both expression are of same value.
                y >= x          both of the function may be same or y may have greater value than that of value of the x.
                y <= x          both of the function are either of same value or y have less value than that of value of x.

Sometimes equality is also included with inequality. For example: Inequality       y >= - x + 1.
Just for example we can draw an inequality y>= (2/3) x - 4 on the plane as:                                                  

The graph shows the equations as per the inequality where they are true on the number line and what values they can grab.(want to Learn more about inequality, click here),
To solve any inequality and get exact solution, we go through the graphing problem on number line. Graphing inequalities problem is the best way to solve any of the inequality. In inequality having one unknown, there may be more than one possible solution (sometimes may be infinite) for a particular inequality. Solving any of the linear inequalities involves the finding of solutions of expressions where variables are not equal on the number line. To solve any inequality we have to graph the inequality on the number line, it is similar to the graphing of linear functions.
This is all about the graphing inequalities problems and if anyone want to know about Graph and Slope of Lines in Grade VIII then they can refer to Internet and text books for understanding it more precisely. Read more maths topics of different grades such as Multistep problems in the next session here.    

Linear Equations VII Grade

Hello my dear students, today we will focus on few more topics of algebra that you study in class VII. As you all now algebra is an interesting subject if you study it with interest and one of the boring subject if things go out of your mind then it is one of the difficult subject. In this article we will talk about Linear/non-linear functions, equations, inequalities. What do you understand by linear function? A linear function can be defined as a polynomial function of single degree. In this we relate a dependent variable with an independent variable in a simple way. It is that mathematical equation in which there is no independent variable that is raised to power greater than one. We can define the simple linear function as a linear equation which is having a single independent variable (y = a + bx), which results in a straight line when we plot the same on the graph.  

Now, let’s switch gear towards the form of linear function. In order to define the linear function we need an equation of first degree. The general form of linear function of first degree is given as:

f= (x, y)/ y = mx + b

In this m and b are constants, x and y are the linear function. When we plot the graph of same it will give a straight line. There are three main forms or types in linear functions.

1. Slope- intercept form: its general form is given as: y = mx + b,

 

2) Point slope for, its general form is given by m = (y –y1) / (x – x1)

3) General Form is given as: Ax + By  + C = 0.

Now, have a look on how to solve the linear functions. Solving the given function by altering its position and simplifying the equation to get Y is generally known as solving linear equations. The solution of these equations provides solutions of the corresponding practical problems. Here are few steps that you can use when you deal with the linear function in word problem form:

1. Read the given problem properly and then note down what is given to us in problem and what other things are required.
2. Denote the unknown quantities by literals like x, y, z, u, v, etc.
3. Translate the statements of the problem step by step in mathematical form so that we can get what to do.
4. After that, look for the quantities that are equal. And then make the equations corresponding to these equality relations.
5. Use an appropriate method and solve the equation formed in step 4.
6. It is the last step and in this you have to check the solution of the problem by substituting the value of the unknown found in step 5. Now, let’s have few examples of linear functions so that you can easily understand the concept and process of solving linear function problems.  

Example 1: Solve for a and b, where, b = a+ 13 and 2a ? b ? 10 = 0

Solution: in this we can easily put the value of the b as b = a + 13 in the equation

      2a ? b ? 10 = 0  

 2a ? (a + 13) ? 10 = 0,

a ? 23 = 0,

a = 23.

Now put the value of a  23 in 2a ? b ? 10 = ,

? y + 36 = 0,

? y = ? 36,

y = 36.

After linear function it’s the time to move towards the next topic that is called as the non- linear function. In simple terms we can define non-linear function as an equation whose graph is not linear and an equation with degree two or more than two. The general form of the same is given as:

f(x) = +….+ a₁x¹ + a₀ where a₀, a₁ ..a_n are stables. In the non linear function a_n is defined as a primary co-efficient and a_n x_n is principal term. The greatest degree of non-linear function is greater than two or similar to two. In non linear function, graph can be a curve, zig- zag line, or any even shape but it can’t be a straight line. The Non linear function can be defined in quadratic form, exponential form and logarithmic form.

Different types of non linear function deals with solving several types of polynomial function where as polynomial function is also called as non linear function.  We can solve the non linear function using substitution method or quadratic equation operations, etc. 

 

Non linear functions play an important role in algebra. Any of the function is not a linear function and can’t be a complete linear function by transforming the Y variable.  

Normally, there are three types of non linear function that we use in mathematics, they are given as:

1. Exponential function
2. Quadratic function
3. Logarithmic function

This is the time to see few examples of the non linear functions.

Example 1: x² - x – 12 = 0, calculate the value for ‘x’ for the given non linear function.

Answer:

As you all know the given equation is in form of x² – 4x +3x - 12 = 0, which resembles with the general form of quadratic equation.

Now, we can get the value for x from the primary term and 3 from secondary term.

x (x – 4) + 3(x - 4) = 0

Now combine the like term (x -4)

(x + 3) (x - 4) = 0

To get the value for x we can associate the factor to zero

a + 3 = 0 or a – 4 = 0

a = -3 or a = 4

a = 4

Thus, the factors a₁ and a₂ are -3, 4.

Example 2:

p² - 3p – 10 = 0, determine the value for p for the non linear function.

Answer:

Now, we can find the factor for the given quadratic equation

p² – 5p +2p - 10 = 0

Now, get the value for x from the primary term and 3 from secondary term.

p(p – 5) + 2(p - 5) = 0

Now we can combine the similar term (p - 5)

(p + 2) (p - 5) = 0

To get the value for x we can associate the factor to zero

p + 2 = 0 or p – 5 = 0

p = -2 or p = 5

Thus, the factors p₁ and p₂ are -2, 5.

Example 3:

x² - 6x + 5 = 0, get  the value of x for the given non linear functions

Solution:

First we can learn the factors for the given quadratic equation

Sum of the roots (-6) = (-5) + (– 1)

Product of roots (5) = (-5) × (-1)

By combining these two roots we can obtain the factor form as,

x² – 5x - x + 5 = 0

Now get the value for x from the primary term and 5 from secondary term.

x (x – 5) - 1(x - 5) = 0

Now we can combine the similar term (x - 5)

(x - 5) (x - 1) = 0

To get the value for x we can associate the factor to zero

x - 5 = 0 or x – 1 = 0

x = 5 or x = 1

Thus, the factors are x1 and x2 and the values are 5, 1.

Now, switch to other topics Equations and inequalities. You all are familiar with both these terms. Here, I will give a brief introduction of both the topics as you are reading the same from earlier classes.  Equations are those mathematical statements that are joined together with help of equal to symbol. There are many different types of equations like linear equations, quadratic equations, polynomial equations, and many other types of equations. There are several different types of methods that you can use to solve different types of equations like linear equations can be solved by simple motion of the variables and constants.

Inequalities can be defined as an expression which is defined using special symbols.  Mainly four different inequality symbols are used using which inequalities are defined like: greater than (>), greater than equal to (<), less than (>=), less than equal to (<=). Solving linear equalities is very simple and you can easily solve them. Solving linear equalities are very similar to solving a linear equation. Inequalities give infinite number of solutions and the answer is right only when the inequality is true.

In all the above mentioned topics for any problem you can take online help and solve all your problems.       

Linear/non-linear functions, equations, inequalities for Grade VII

Linear functions: 

Functions which have x as the input variable, where x has an exponent of only 1.

For example:
y = mx + c
Here we can see that x has an exponent of 1 in each equation.

The above functions such as this yields graphs that is straight line, and, thus, the name linear is alloted to it.

 

Linear functions are of three categories:

1) Slope-Intercept Form

2) Point-Slope Form

3) General Form

 

1) Slope-Intercept Form

A simple way to define a linear function is through the slope-intercept form of a linear function. When drawn on a common (x, y) graph it is given as:

y = mx + b

Or, in a formal function definition:

f(x) = mx + b

 

Basically, this function describes a set, or locus, of (x, y) points, and each of these points are located in a straight line. 'm' represents the slope of the line. 'b' represents the y-coordinate for the spot where the line crosses the y-axis and this point is called the 'y-intercept'.

 

2) Point-Slope Form

In the 'point-slope form' for the equation of a line, the definition does not start with 'y =' or 'f(x)=', so it's not written in a common function definition form.

But it can be very well written in algebra:

y = m(x - x1) + y1

and as a functional definition it can be written as:
f(x) = m(x - x1) + y1

To conclude the equation with the given slopes and points, we can say that the variable m is the slope of the line and the point x1, y1 are the points on the line. If anyone knows the slope of and the coordinates for one point on the line, then the he or she can include or can enter those values into this equation, and the equation would then define a set, or locus, of all the points on that line.

 

3) General Form

Let's say A, B, and C are three numbers forming an equation where the same equation can be assumed as:

0 = 2x + 3y + 4

 

All the points with (x, y) coordinates that can make the above statement true forms a line.

 

Using algebra, this general form can be changed into a slope-intercept form, and then you would know the slope and y-intercept for the line.

 

Non-linear Functions:

 

A non-linear function is defined as a polynomial function of two or even higher degree. The linear function is a polynomial function of degree 1. A quadratic function is a polynomial function of degree 2, defined by an equation of the form

y = ax² + bx + c

The degree of a polynomial function is the degree of the polynomial itself.

For example:

Let us find the degree of the following function and also the points where x and y intercepts.

y= x²-2x-3

The degree of the given equation is 2. To find the point where its graph cuts the x axis, we can make y = 0. now we get,

x²-2x-3 = 0

Solving the equation we get x = 3 and x = -1. These are points where the function graph intercepts the x-axis.
To find the point where the graph intercepts the y-axis, assume x = 0. Now we get y = -3.

 

 

Linear Equations:

 

A linear equation appears like any other form of equation. A linear equation is made up of two expressions sets that are equal to each other. A linear equation is different as, it can have one or even two variables.

No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction. Linear equations graph as straight lines.

A linear equation in two variables shows an experiment between two variables in which the value of one of the variables depends on the value of the other variable. In a linear equation in x and y, x is called an independent variable and y depends on it. We call y the dependent variable. If the variables have other names, yet do have a dependent relationship. The independent variable is plotted along the horizontal axis. Most linear equations are functions (that is, for every value of x, there is only one corresponding value of y). When you assign a value to the independent variable, x, you can compute the value of the dependent variable, y. You can then plot the points named by each (x,y) pair on a coordinate grid.

 

Lines with the Same Slope

Lines with the same slope are either the same line, or parallel lines.

 

Consider 2x + y – 6 = 0. This equation is not in slope-intercept form. There are two ways to put it in slope-intercept form.

1. The actual equation is 2x + y – 6= 0

Subtract y from each side. 2x + y – y – 6 = 0 – y

2x – 6 = 0 – y

Multiply each side by 1.

 

1(2x – 6)= 1(y)

2x + 6 = y

 

2. Show the original equation. 2x + y – 6 = 0

Add 6 to each side. 2x + y – 6 + 6= 0 + 6

2x + y = 6

Subtract 2x from each side. 2x – 2x + y = 6 – 2x

y = 6 – 2x

The given two equations, 2x + 6 = y and y = 6 – 2x are equal because you can change one equation into the other by using the symmetric property of equality, which states that if a = b, then b = a and the commutative property, which states that a + b = b + a.

commutative property 2x + 6 = y 6 – 2x = y

symmetric property 6 – 2x = y y = 6 – 2x

 

Nonlinear Equations:

Equation whose graph does not form a straight line (linear) is called a Nonlinear Equation. In a nonlinear equation, the variables are either of degree greater than 1 or less than 1, but never 1.

For example:

7x⁵+ y = 0 and x³+ 13x²-4xy² = 0 are the examples of nonlinear equations. We can also consider that the form x^1/3 + y^1/3 = 0 is a nonlinear equation.

Lets take an example of an equation:
(x+ 2)²= 6

This is a non-linear equation because, as per the definition, an algebraic equation is said to be linear if the variable or variables in the equation are of first degree.
Considering the equation (x+ 2)² = 6

We can see that x can be concluded as x raised to the power 2 or x².
Therefore, (x+ 2)²= 6 is not a linear equation.

Linear inequalities:

Phenomena of linear inequalities is a set of linear inequalities that anyone comes across all in the same time. Normally, a person starts with two or three linear inequalities.

 

For example:

• Solve the following system:

  2x– 3y < 12
  x + 5y < 20
  x > 0

  It is better to solve a large number of linear inequalities for "y" on one side. Solving the first two inequalities, get the rearranged system as:

  y > ( ²/₃ )x – 4
  y < ( – ¹/₅ )x + 4
  x > 0

Nonlinear Inequalities:

To solve non linear inequalities, We can use the following steps the following steps:

1.Place every component on the left hand side so that we have for example equation is greater than (>) 0.

2.Factorize and set equal to zero.

3.Solve and put the answers on the number line. Doing so it will divide the number line into two or three parts.

4.Select the value of each part and analyze them for each region and put that acquired value into each of the factors. Put '+' or '-' over the region looking into the values decide positive or negative values.

5.If the region has two '+' or two '-' then the region is positive. If the region has one of each then the region is negative.

6.If the inequality is "<" then include the negative regions.

7.If the inequality is ">" then include the positive regions.

8.If the inequality is a less than or greater than or equal to then include the endpoints with solid dot and the interval []

9.If the inequality is a less (greater) than then do not include the endpoints by showing an open dot and the interval () Lets take an example, Solve x2 + 3x > -2 1.x2 + 3x + 2 > 0 2.(x + 2)(x + 1) = 0 3.x = -2 or x = -

1 This divides the number line into three regions.

4.For the left region we choose-5 and have, -5 + 2 < 0, -5 + 1 < 0 5.For the middle region we choose -1.5, and get, -1.5 + 2 > 0, -1.5 + 1 < 0 For the left region we choose 0 and get, 0 + 2 >0, 0+ 1 >0 6.

The left and right regions are positive and the middle region is negative. 7.The inequality acquired is a greater in quantity we have the solution: (-,-2) U (-1,)