How to solve inequalities and graph its solution?

Question

Solve the inequality $ \begin{vmatrix} \frac{2}{x-1} \end{vmatrix}>5 $ and graph its solution.

• What do you mean by inequalities?
• How do you prove an inequality?
• How to graph the solution?

Sorry if I am asking something that is too obvious. I didn't find a solution for this.

Answer 1

You can think of an inequality as an equation only with the $=$ sign replaced by one of the following : $<,>,\leq,\geq$

Here are some rules:

1.) Generally speaking, you solve them the same way you solve regular equations (by performing the same operation on both sides). The exception is if you multiply or divide by a negative number, the direction of the inequality is reversed.

2.) $|a| < b$ means $-b < a$ and $a < b$

3.) $|a| > b$ means $a > b$ or $a < -b$

Answer 2

Hint. Square both sides. This gives $$4>5^2(x-1)^2.$$ Then factorise to get $2^2-(5(x-1))^2>0,$ or $$(2- 5(x-1))(2+5(x-1))>0.$$ Simplify the factors, then draw a graph.