Question

How do you solve inequalities with fractions ?

Answer 1

Here we are asked to solve the inequalities with fractions. Firstly, we must know which basic mathematical operations have to apply to simplify it. Here we make use of mathematical operations such addition, subtraction, multiplication to solve the given inequality. We first try to remove the fractions. Then we try to bring the like terms together to make the process easy. Then we keep unknown terms on one side of the inequality and the constant term on the other side. Afterwards we solve it to obtain the desired result.

Complete step by step solution:
We need to mention the procedure to solve the inequalities with fractions. We try to explain this by the following steps. Even though the inequalities with fractions seem to be difficult, if we follow the below steps they can be solved easily.
The steps are as follows :
(1) Before beginning to use any processes to try and solve the inequality, firstly begin by simply taking the inequality before you. Make note of any negatives that you will need to remember to carry through while solving the problem.
(2) Here notice all the mathematical operations in the inequality such as multiplication, subtraction, exponents, parentheses and such.
(3) Use the order of operation in reverse to begin to solve the problem. Actually we need to begin with parentheses, exponents, multiplication or division, addition or subtraction. But when we are solving for a variable, we will use the order of the operations in reverse.
(4) Write the inequality in the correct form. One side must be zero and the other side must contain only the fractions. In case if there are more than one fraction, then simplify it by multiplying all terms by the least common denominator of all fractions.
(5) Simplify by combining the like terms on the left hand side.
(6) Then add or subtract the quantities to obtain the unknown on one side and the constant terms on the other side and then simplify the inequality.
(7) Remember that if in the problem, we have to multiply or divide by a negative number, then we need to flip the sign of the inequality.

Note:
To understand these steps properly, let us consider one example.
Let $4 < \left( {\dfrac{x}{8}} \right) + 5$.
Now we subtract 5 from both sides, we get,
$\Rightarrow 4 - 5 < \left( {\dfrac{x}{8}} \right) + 5 - 5$
$\Rightarrow - 1 < \left( {\dfrac{x}{8}} \right) + 0$
$\Rightarrow - 1 < \left( {\dfrac{x}{8}} \right)$
Now multiply both sides by 8, we get,
$\Rightarrow - 1 \times 8 < 8 \times \left( {\dfrac{x}{8}} \right)$
$\Rightarrow - 8 < x$
Hence the solution for $4 < \left( {\dfrac{x}{8}} \right) + 5$ is $x > - 8.$