Rational inequalities 

A rational inequality is an inequality that contains a rational expression, where a rational expression is a ratio of two polynomials. Solving a rational inequality is somehow similar to finding solutions to linear inequalities. Here, we must remember when we multiply or divide by a negative number, the inequality sign must reverse.

One simple method would be to assign value to the variable in the rational inequality and see if the outcome or solution satisfies the inequality. (meaning values that satisfy the inequality). A simple breakdown of the steps would be:

  • Rearrange the rational inequality in such a way that a single fraction appears on one side and then proceed on to make one side of the inequality as equal to zero.

for example,

Take x-3/x + 5 < 2 

Subtract 2 from both sides x-3x + 5 – 2 < 0

x-3/x+5 – 2 * x+5x+5 < 0

x-3-2x+10x+5 < 0

-x+7/x+5 < 0

  • Find the x-value(s) that make the numerator equal to zero.

            – x + 7 = 0  ⇒  x = 7

  • Find the x-value(s) that make the denominator equal to zero.

            x + 5 = 0    ⇒   x = -5

  • These two values namely 7 and -5, derived from the two steps, will determine the number line test intervals.
  • Mark the values (7 and -5) on a number line so that intervals are created on the line. 
  • Let us now see by picking a point (number) within each interval and evaluate whether this number or point actually satisfies the inequality. 

Test point = −6 

-6-3/-6+5 < 2

-9-1 < 2 (which is false)

Test point = 2

2-3/2+5 < 2

-17 < 2 (which is true)

Test point = 8

8-3/8+5 < 2

5/13 < 2 (which is true)

  • As an inequality, the solution is x > − 5 or x > 7
  • Stated in interval, the solution is (−5, + 7) and (7, + ∞)

Example:3x-10/x-4 > 2

3x-10/x-4 – 2 > 0

3x-10/x-4 – 2

3x-10/x-4  – 2x – 4/x-4 > 2 (Multiplying 2 by (x−4)/(x−4))

3x-10/-2(x-4)x-4 > 0 (Bringing it together since they have a common denominator)

x-2x-4 > 0  (Simplify)

Let us find ‘points of interest.’ 

When x = 2, we have  0x-4   > 0 , which is ‘=0’

When x = 2, we have 0x-4 >0

0 / x − 4 > 0, which is ‘=0’

When x = 4, we have x – 2 / 0 > 0, which is undefined.

Graphing Linear Inequalities 

If the inequality involves either < or >, the dotted lines on the graph will indicate whether they belong to the solution set. If they include ≤ or ≥, the lines will be dark, indicating that they belong to the solution set. If linear inequalities in one variable are plotted on a number line, then solving the output by finding the value of the variable will give solutions. That is why it makes sense to use only a number line graph rather than cartesian plane for solving linear inequalities in one variable.

Summary

Linear inequality can form a crucial part of rational inequality. If we know how to solve a linear inequality, we know how to solve a rational inequality. Rational inequalities are an inequality that contains a rational expression or equation. The best way to solve this type of equation is to eliminate all the denominators using the least common denominator. Four operations on linear inequalities are addition, subtraction, multiplication, and division. Lastly, we must remember that when we multiply or divide by a negative number, the inequality sign reverses.

Linear inequalities in one variable

In Mathematics, If an expression equates to two expressions or values, it is called an equation. But if it relates to two expressions or values with a ‘>’ (greater than), ‘≤’ (less than or equal), and ‘≥’ (greater than or equal), it represents a linear inequality.   

A linear inequality contains one of the symbols of inequality such as < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), and ≠ (not equal to). A linear inequality would seem like a linear equation, with the inequality symbol replacing the equality sign.

Examples: x – 5 > 3x – 10, 5x + 27>0, 20x – 7 ≥ 0

How to solve inequalities?

  • Write the inequality as an equation.
  • Solve the equation for single or more values. 
  • Assign all the values on a number line.
  • Try to assign all excluded values on the number line using open circles.
  • Mark the intervals.
  • Pick any number from each of these intervals and now assign this value in the first inequality and find out whether this number satisfies the inequality.
  • These intervals which are satisfied are the solutions to the problem.

Example: x + 3 < 7

                x + 3 /− 3 < 7 − 3 (Minus 3 from both sides) 

                x < 4 

Our solution is: x < 4

So, we can say x can be any value less than 4.

Some important things about inequalities:

  • Many simple inequalities can be solved by adding, subtracting, multiplying, or dividing both sides until you are left with the variable on its own.
  • Don’t multiply or divide by a variable.
  • If you multiply or divide both sides with a negative number, it will reverse the inequality sign.
  • If you swap left and right-hand sides, it will also change the direction of the inequality.
  • If we have strictly less than or greater than the symbol, we never get any closed interval in the solution.
  • We always get open intervals at ∞ or -∞ symbols because they are not included numbers.
  • Write open intervals always at excluded values when solving rational inequalities.
  • Excluded values should be taken care of only in case of rational inequalities.