## Inequalities involving logarithm

Students often ask why sometimes need to flip the inequality sign when dealing with logarithm.

This is a question asked by a student:  For 0<x<1, both ln x and lg x are negative. When we divide by negative number, we need to change the sign.

For x>1, both ln x and lg x are positive. There is no need to change the sign when divide by ln x or lg x.

## Challenging Inequality problem 2

Given that k>0, solve the inequality Solution

First, we determine the roots of  Then we need to determine whether the roots are bigger or smaller than k, so we can sketch the number line.

Since k> 0  ## Challenging Inequality problem 1

Solve the inequality From G.C, 0.5<x< 2.81, x not equal to 2 (This is the easy part)

Hence solve This is the HOT (Higher Order Thinking) part.

Method 1

Let f(x) = Therefore f(x+2) = f(x+2) is translation of f(x) by 2 units in the negative X direction.

Therefore, the solution is -1.5< x<0.81, x not equal to 0

This solution combines the concept of inequality and transformation of graph.

Method 2

Replace x by x+2.

Therefore 0.5<x+2< 2.81, x+2 not equal to 2

-1.5<x<0.81, x not equal to 0

## Inequalities: Use And/OR correctly

Many students use the words “And/Or” indiscriminately in solving inequalities. However, there is a very important difference. ‘And‘ means you take the intersection of the solutions sets. ‘Or‘ means you take the union of the solutions sets.

Example 