Question

How do you solve the inequality \[\dfrac{x}{{x - 1}} < 1\] ?

Answer 1

Hint: An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. Here we need to solve for ‘x’ which is a variable. Solving the given inequality is very like solving equations and we do most of the same thing but we must pay attention to the direction of inequality \[( \leqslant , > )\] . We have a simple linear equation type inequality and we can solve this easily.

Complete step-by-step answer:
Given \[\dfrac{x}{{x - 1}} < 1\]
We need to solve for ‘x’.
Since we know that the direction of inequality doesn’t change if we multiply a number on both sides. We multiply \[x - 1\] on both sides of the inequality we have,
 \[\dfrac{x}{{x - 1}} < 1\]
 \[x < x - 1\]
We subtract ‘x’ on both sides we have,
 \[x - x < - 1\]
 \[0 < - 1\] .
That is we have no solution. The given inequality is inconsistent.

Note: We know that \[a \ne b\] is says that ‘a’ is not equal to ‘b’. \[a > b\] means that ‘a’ is less than ‘b’. \[a < b\] means that ‘a’ is greater than ‘b’. These two are known as strict inequality. \[a \geqslant b\] means that ‘a’ is less than or equal to ‘b’. \[a \leqslant b\] means that ‘a’ is greater than or equal to ‘b’.
The direction of inequality do not change in these cases:
 \[ \bullet \] Add or subtract a number from both sides.
 \[ \bullet \] Multiply or divide both sides by a positive number.
 \[ \bullet \] Simplify a side.
The direction of the inequality change in these cases:
 \[ \bullet \] Multiply or divide both sides by a negative number.
 \[ \bullet \] Swapping left and right hand sides.