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Inequalities with Fractions

Solving inequalities with fraction is quite similar to solving the multi step equations with fractions as coefficients. The properties of inequality need to be followed while solving inequality. 


Here is an example of  linear inequality  with integer coefficients .

$4x - 6 < 12$

Solve using rules of solving the equation. All the steps will be similar except the symbol of = which will be replaced by given inequality. (< for less than , > for greater than ,$\leq$  for less than or equal to and $\geq$  for greater than or equal to)

Step 1: Isolate the variable

Add 6 on both sides

4x - 6 + 6 < 12 + 6

4x < 18

Divide both sides by 4

$\frac{4x}{4}$  < $\frac{18}{4}$

$x < 4.5$

The solution can be written in interval notation as (-$\infty$, 4.5).

On number line :

Inequalities with Fractions

Inequalities with Fractions

The best way is to get rid of fractions in solving inequalities with fractions. The first step should always be to isolate the variable. Convert all the fractions involved to like fractions by taking the LCD and using appropriate multiples. 

Once the variable is isolated, use proper multiplication by reciprocal of the coefficient of the variable to get the solution.  

Point to remember: The inequality sign reverses when both the sides of the given inequality are either multiplied or divided by a negative number.

Related Calculators
Calculator for Inequalities Inequality Calculator
Mixed Fraction to Improper Fraction Compound Inequalities Calculator
 

Examples

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Example 1: Solve 

$\frac{1}{3}$ (x-2) > $\frac{7}{2}$

Solution:

Multiply all the terms by 3

$\frac{1}{3}$ $\times$ 3(x-2) > $\frac{7}{2}$ $\times$ 3

(x-2) > $\frac{21}{2}$

Add 2 on both sides

x  > $\frac{21}{2}$ + 2

 x > $\frac{25}{2}$

Check the solution , first check the end point x = $\frac{25}{2}$ in the related equation

$\frac{1}{3}$ ($\frac{25}{2}$ - 2) = $\frac{7}{2}$

$\frac{1}{3}$ ($\frac{25}{2}$ - $\frac{4}{2}$) = $\frac{7}{2}$

$\frac{1}{3}$ ($\frac{21}{2}$) = $\frac{7}{2}$

Simplifying the left side ,we get

$\frac{7}{2}$ = $\frac{7}{2}$

Now, check any value for $x$ in the region  $x >$ $\frac{25}{2}$

Let $x = 14$

$\frac{1}{3}$ (14-2) > $\frac{7}{2}$

$\frac{1}{3}$ $\times$ (12) > $\frac{7}{2}$

Simplify the left side. 

4 > $\frac{7}{2}$

which is true ,since $\frac{7}{2}$  > $\frac{8}{2}$  

Answer : x > $\frac{25}{2}$
Example 2:   Solve

$\frac{1}{3}$ x - $\frac{5}{6}$ > $\frac{-5}{2}$

Solution:

Add $\frac{5}{6}$ on both sides

$\frac{1}{3}$ $x$ - $\frac{5}{6}$ + $\frac{5}{6}$ > $\frac{-5}{2}$ + $\frac{5}{6}$

$\frac{1}{3}x $ > $\frac{-5}{2}$ + $\frac{5}{6}$

LCD of 3,6 and 2 is 6. Use appropriate multiple to get like fractions.

$\frac{1}{3}$ $x$ $\times$ $\frac{2}{2}$ > $\frac{-5}{2}$ $\times$ $\frac{3}{3}$ + $\frac{5}{6}$

$\frac{2}{6}$ $x$ > $\frac{-15}{6}$ + $\frac{5}{6}$

$\frac{2}{6}$ $x$ > $\frac{-15+5}{6}$

$\frac{2}{6}$ $x$ > $\frac{-10}{6}$

Multiply both sides by  $\frac{6}{2}$  to isolate $x$ 

$\frac{2}{6}$ $x$ $\times$ $\frac{6}{2}$ > $\frac{-10}{6}$ $\times$ $\frac{6}{2}$

x > $\frac{-10}{2}$

x > -5

Inequalities with Fractions Examples

Check the solution ,plug in x = -5 in related equation(replace the inequality sign by =)

$\frac{1}{3}$ (-5) - $\frac{5}{6}$ = $\frac{-5}{2}$

$\frac{-5}{3}$ - $\frac{5}{6}$ = $\frac{-5}{2}$

Take LCD on left side = 6

$\frac{-5}{3}$ $\times$ $\frac{2}{2}$ - $\frac{5}{6}$ = $\frac{-5}{2}$

$\frac{-10}{6}$ - $\frac{5}{6}$ = $\frac{-5}{2}$

$\frac{-10}{-5}$ {6} = $\frac{-5}{2}$

$\frac{-15}{6}$ = $\frac{-5}{2}$

Divide the fraction on left side by $\frac{3}{3}$ to reduce it 

$\frac{-15}{6}$ $\times$ $\frac{3}{3}$ = $\frac{-5}{2}$

$\frac{-5}{2}$ = $\frac{-5}{2}$

Now, check for the value of x in inequality region. Say,x = 0

$\frac{1}{3}$ (0) - $\frac{5}{6}$ > $\frac{-5}{2}$

-$\frac{5}{6}$ > $\frac{-5}{2}$

This statement is true .So,

Answer  x > -5
Example 3 :  

$\frac{3}{4}$ - $\frac{5}{8}$
 x $\geq$  2

Solution: 

Subtract $\frac{3}{4}$ from both sides

$\frac{3}{4}$ - $\frac{3}{4}$ - $\frac{5}{8}$ x $\geq$  2 - $\frac{3}{4}$

-$\frac{5}{8}$ x $\geq$  2 $\times$ $\frac{4}{4}$ - $\frac{3}{4}$

-$\frac{5}{8}$ x $\geq$  $\frac{8}{4}$ - $\frac{3}{4}$

-$\frac{5}{8}$ x $\geq$  $\frac{8-3}{4}$

-$\frac{5}{8}$ x $\geq$  $\frac{5}{4}$ 

Multiply both sides by reciprocal of  $\frac{5}{8}$

-$\frac{5}{8}$ $\times$ $\frac{8}{5}$ x $\geq$  $\frac{5}{4}$ $\times$ $\frac{8}{5}$

$-x$ $\geq$ {2}

Divide both sides by -1.

As per rule of inequality, the inequality sign reverses when both the sides are multiplied by a negative number.

-$\frac{x }{-1}$  $\leq $ $\frac{2}{-1}$

x $\leq $ -2.

Check  with $x = -2$  .

$\frac{3}{4}$ - $\frac{5}{8}$(-2) = 2

$\frac{3}{4}$ + $\frac{5}{4}$ = 2

$\frac{8}{4}$ $= 2$

$2 = 2$ , which is true.

Answer: $x \leq  -2$

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