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Topic 3: Equations and Inequalities
 

 

1. EQUATIONS OF THE FIRST DEGREE

As the name suggests, an equation always contains an equal sign. A mathematical sentence containing two expressions connected by the symbol “=” means “is equal” is called an equation.

1.1. Simple equations

If the highest power in equation is 1 then the equation is said to be the equation of the first degree. Consider the statement . This statement is neither true nor false until we choose a value for x. If , the statement is true. So  is called the solution of the statement and the solution set is . The number 3 is called the root of the equation. Thus, to find a solution to the given equation is to find the value that satisfy that equation.

Example

Solve, in set of real numbers, the following equations:

1)

2)

3)

Solution

1)

   

2)

  

3)

  

1.2. Equations product/quotients

When we are given the equation of the form AB=0 we use the fact that if AB=0 then A=0 or B=0. Also by cross product theorem, if .

Example

Solve, in set of real numbers, the following equations

1)

2)

3)

4)

Solution

1)

   

2)

   

 3)

    

4)

    

2. INEQUALITIES OF THE FIRST DEGREE

As the name suggests, inequality always contains an inequality sign. The symbols used are the following:

“<”: less than

“>”: greater than

”: less than or equal to

”: greater than or equal to

2.1. Simple inequalities

The statement is true only when . If x is replaced by 10, we have a statement which is false. To be true we may say that 3 + 2 is less than 10 or in symbol . In this case we no longer have equality but inequality.

Suppose that we have the inequality , in this case we have an inequality with one unknown. Here the real value of x satisfies this inequality is not unique. For example 0 is a solution but 2 is also a solution. In general all real numbers less than 6 are solutions. In this case we will have many solutions combined in an interval.

Open interval

If a and b are two real numbers such that then the set of numbers between a and b (excluding a and b) is called an open interval, denoted by or .

Closed interval

The set  consisting of all real numbers between a and b (including a and b) is called a closed interval and is denoted by .

Half-closed or half-open interval

The interval  is half-open on the left (or half-closed on the right).
The interval  is half-closed on the left (or half-open on the right).

Now, the solution set of the inequality  is an open interval containing all real numbers less than 6 (excluded). How? We solve this inequality as follow:



And then

Notice

Remember that when the same real number is added or subtracted from each side of inequality the direction of inequality is not changed. Also, the direction of the inequality is not changed if both sides are multiplied or divided by the same positive real number but the direction of the inequality is reversed if both sides are multiplied or divided by the same negative real number.

Example

Solve in set of real numbers

1)            2)          

3)          4)

Solution

1)   

 

 

2)  

 

3)

Since any real number times zero is zero and zero is greater than -18, then the solution set is the set of real numbers.

 

4)

Since any real number times zero is zero and zero is not less or equal to -1 then the solution set is the empty set.

2.2. Inequalities products / quotients

To solve the inequality of the form we follow the following steps:

1. First we solve for

2. We construct the table called sign table, find the sign of each factor and then the sign of the product or quotient if we are given a quotient. For the quotient the value that makes the denominator to be zero is always excluded in the solution. For that value we use the symbol || in the row of quotient sign.

3. Write the interval considering the given inequality sign.

Example

Solve in set of real numbers the following inequalities

1)

2)

Solution

1)

The next is sign table

Since the inequality is we will take the interval where the product is negative. Thus,

2)

3. EQUATIONS OF SECOND DEGREE

3.1. Quadratic equations

Equation of second degree has the form . To solve this equation, we use the following methods:

Factorization method:

We factorize the the first expression (See Factorization of Polynomials) and we use the fact that if AB=0 then A=0 or B=0

Example

Solve in set of real number

1)                                      2)

Solution

1)

 

2)

The expression can not be factorized. Then the solution set is empty set.

Discriminant method

Consider the equation

where

  • If , there are two distinct real roots,
  • If  , there is a signel double root
  • If , there is no real root.

Example

Solve

1.                   2.                    3.

Solution

1.

2.

3.

3.2. Equations reducible to quadratic form

Irrational Equations

Irrational equation is an equation involving radicals.

To solve an irrational equation, follow these steps:

  • Isolate a radical in one of the two members and pass it to another member of the other terms which are also radical.
  • Square both members.
  • Solve the equation obtained.
  • Check if the solutions obtained verify the initial equation.
  • If the equation has several radicals, repeat the first two steps of the process to remove all of them.

Example

Solve

Solution

For

Biquadratic equations

Biquadratic equations are the equations that if we look at them in the correct light we can make them look like quadratic equations. The general form of a biquadratic equation is

To solve this equation, we put

Example

Solve

Solution

4. INEQUALITIES OF THE SECOND DEGREE

To solve inequality of the second degree like , we transform the left hand side into factor form and we make a sign table using the obtained factors.

Example

Solve

Solution

Example

Solution

5. SYSTEM OF LINEAR EQUATIONS

Consider two equations and . These two equations represent two lines in cartesian system. The point of intersection of those lines is the solution of the following system

This syatem can be solved algebrically (addition method, substitution method or Cramer's rule) or graphically.

Addition (Combination or Elimination) method

We try to combine the two equations such that we will remain with one equation with one unknown. We find two numbers to be multiplied on each equation and then add up such that one unknown becomes zero

Example

1)                2) 

Solution

1)

2)

Substitution method

We find the value of one unknown in one equation and put it in another equation to find the value of the remaining unknown.

Example

1)                                       2)

Solution

1)

2)

There is infinity number of solutions.

Cramer's rule

Consider the following system

To use Cramer's rule , we first check if the unknown are in the same position.

We first find

Example

1)                 2)                                 3)

Solution

1)

2)

Impossible. There is no solution.

3)

Indeterminate. There is infinity number of solutions.

Grapical method

To solve the system of two linear equations, graphically, follow the following steps:

  • Find at least two points for each equation.
  • Plot the obtained points in xy plane and join these points to obtain the lines. Two points for each equation give one line.
  • The point of intersection for two lines is the solution for the given system

Example

Solve the following system by graphical method

1)                              2)                                   3)

Solution

1)

Graph

From the graph

2)

We see that the two lines are parallel and do not intersect. Therefore there is no solution. Note that the gradients of the two lines are the same.

3)

We see that the two lines coincide as a single line. In such case there is infinite number of solutions.

6. SYSTEM OF TWO LINEAR INEQUALITIES

An inequality with two unknowns has a range of solutions. This range can be shown on a graph by a region with boundaries.
To represent a linear inequality on a graph:

  • Draw the boundary line
  • Identify which side of the line contains solutions to the inequality and whether the boundary line is included.
  • Shade the area that does not contain the solutions.

For inequality , we draw solid boundary lines and for inequality < or >, we draw dashed boundary lines. A dashed boundary line indicates that points along the line are not part of the solution. A solid boundary line indicates that the points along the line are part of the solution.

Example

Show, on a graph, the region defined by:

Solution

7. APPLICATIONS

 

 
 

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