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Solving Inequalities with Fractions

The general rules for solving inequalities are also applied to solving inequalities with fractions. The steps to be followed or the operations to be performed vary according to the difficulty added by fractions in solving inequalities. Let us learn the different strategies applied by solving some examples.

Solving by Eliminating the Fractions:
Inequalities with fractions are generally solved by eliminating the fractions. This is done by multiplying the inequality suitably.
In the case of inequalities containing multiple fractions, the inequality is multiplied by the LCM of the denominators present.

Given below are some of the examples in solving by eliminating the fractions.

Example 1:

Solve $-\frac{4}{7}$ x $\leq$ $\frac{8}{7}$

$7 \times$ $-\frac{4}{7}$ x $\leq$ 7 $\times$ $\frac{8}{7}$ (Inequality is multiplied by 7 to get rid of the denominators)

- 4x $\leq$ 8

$\frac{-4x}{-4}$ $\geq$ $\frac{8}{-4}$ (Inequality is divided by -4 to isolate the variable. The inequality sign is reversed)

x $\geq$ -2       

Example 2:

Solve $\frac{2z-5}{3}$ - $\frac{3z+1}{4}$ $\geq$ $\frac{11}{12}$

Solution:

12[$\frac{2z-5}{3}$ - $\frac{3z+1}{4}$] $\geq$ 12 x $\frac{11}{12}$    (Inequality is multiplied by LCM 12)

4(2z - 5) - 3(3z + 1) $\geq$ 11
8z - 20 - 9z - 3 $\geq$ 11
- z - 23 $\geq$ 11                       (Left side is simplified)
- z - 23 + 23 $\geq$ 11 + 23                                              
-z $\geq$ 34
z $\leq$ - 34        (Solution with inequality symbol reversed)

Solving by Grouping the Fractions:
Given below is an example on solving by grouping the fractions.

Example:

Solve 3x - $\frac{3}{8}$ < $\frac{1}{2}$

Solution:

The above inequality can be solved without eliminating the fraction, but by combining them.

3x - $\frac{3}{8}$ + $\frac{3}{8}$$\frac{1}{2}$ + $\frac{3}{8}$ ($\frac{3}{8}$ is added to isolate the term containing the variable.)

3x < $\frac{1}{2}$ + $\frac{3}{8}$

3x < $\frac{7}{8}$

x < $\frac{7}{24}$ 

Solving by Cross Multiplication:
When the inequality contains isolated terms on sides, the inequality can be solved by cross multiplication.

$\frac{3}{2}$$x$ < - $\frac{2}{5}$

$\frac{3x}{2}$ < $\frac{-2}{5}$     (The fractions are re written keeping the denominators positive)

Solving by Cross Multiplication

(3x) (5) < (-2) (2)    
15x < - 4

x < - $\frac{4}{15}$

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