Irrational equations and Irrational Inequalities




Irrational equations

A diculty in solving irrational equations

Example :
 
    _________
  \/ x + 8     = x + 2
The general method to solve an equation is to replace, in succession, the equation by an equivalent equation.

We tend to square both sides of the equation, but the next equivalence is false.

 
      _________
    \/ x + 8     = x + 2    <=>  x + 8 = (x + 2)2

  Indeed, -4 is solution of

    x + 8 = (x + 2)2

   but it is not a solution of
      _________
    \/ x + 8     = x + 2

   It is obvious that the following expression is correct.
      _________
    \/ x + 8     = x + 2    =>   x + 8 = (x + 2)2
All solutions of the equation on the left side are solutions of the equation on the right side but not vice versa. The equation on the right can have more solutions.

Nevertheless, we will solve irrational equations by squaring both sides. But we must be aware that, by squaring, the new equation can have more solutions than the original one.
At the end, the 'false solutions' are deleted.

There are different ways to find these 'false solutions'. One way is to build suitable inequalities in advance, to find the 'false solutions'. We don't follow this method here.
The simplest way is to test each solution to the given equation. If the given equation is not satisfied for that solution, it is labeled as a 'false solution'.

Let's use this procedure to the equation

 
      _________
    \/ x + 8     = x + 2

=>   x + 8 = (x + 2)2

<=>  ...

<=> x = 1 of x = -4
We test these two values to the given equation. We see that -4 is a false solution and it must be deleted. The only solution is 1.

Some examples

Exercises

 
        _______
  3 + \/ 3x + 1  = x     ( Solution : 8)

    _______      _______
  \/ x + 27  - \/ x - 5  = 2  ( Solution : 54)

    _______      _______
  \/ 7x + 2  - \/ 3x + 1  = 1  ( Solution : 1)

     __________
    /    ______      _______
  \/ 2 \/ x + 1  = \/ 3x - 5   ( Solution : 3)

Extra examples on the net

Irrational Inequalities

Four step method

step 1

Bring all terms of the inequality to the left side. This creates an irrational function f(x) in the left side of the inequality. Find the domain of the function f(x).

step 2

Find the zero's of the irrational function f(x).

step 3

Draw the axis of the real numbers.
Rule out the area that does not belong to the domain of f(x)
Mark the zero's of f(x) on the axis.
Determine the sign of f(x) in all intermediate intervals. This can be done with the aid of the image of a simple x-value.

step 4

From the result of step 3, you can read the solution set of the inequality.

Examples




Topics and Problems

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