Dr. J's Maths.
Where the techniques of Maths
are explained in simple terms.

Algebra - Inequalities - Inequalities - variable in the denominator.
Test Yourself 1 - Solutions.


 

Multiplication can create an x2 term.

Hence the required domain is either:

  • between 5 and 6 or
  • less than 5 and greater than 6

Test a number in the original inequality - 0 is easy and it is in the second domain above. We get 0 after substitution which is less than 3 - so the inequality is not satisfied (sad).

Hence the required domain is from 5 to 6. But the correct domain cannot include 5 because then the denominator would be 0 - not generally a good thing!!!

So our solution is 5 < x ≤ 6.

Hence the required domain is either:

  • between -1 and 2 or
  • less than -1 and greater than 2.

Test a number in the original inequality - 0 is easy and it is in the first domain above. We get 1.5 after substitution which is greater than 1 - so the inequality is satisfied (so be happy).

Hence the required domain is from -1 to 2. The correct domain cannot include 2 because then the denominator would be 0 - so the fraction would be undefined!!!

So our solution is -1 ≤ x < 2.
 
Multiplication can create an term greater than x2.

Hence the required domain is either:

  • between 1 and 2 or greater than 3;
  • less than 1 or between 2 and 3.

Test a number in the original inequality - 0 is easy and it is in the second domain above. We get -1.5 after substitution into the left side which is less than 0 - so the inequality is satisfied (happy).

So our solution is x ≤ 1 or -2 ≤ x ≤ 3.
   
  Where on the graph of y = loge (x-2) is the gradient of the curve greater than 1.