How to Solve Rational Inequalities
Let’s take an example to solve rational inequalities.
Example: Solve the inequality: (x + 3) / (x – 1) < 2
Solution:
Following are the steps to solve inequality:
Step 1: Write the inequality in the form of equation: (x + 3) / (x – 1) < 2
(x + 3) / (x – 1) = 2
Step 2: Solve the equation:
(x + 3) / (x – 1) = 2
(x + 3) = 2(x – 1)
x + 3 = 2x – 2
2x – x = 3 + 2
x = 5
From above step we obtain value x = 5
Step 3: From above values the intervals are (-∞,1), (1, 5), (5, ∞)
Since, the inequality is < that does not include equal to, so we use open bracket for the obtained value.
Since, for x = 1 the inequality is undefined, so we take open bracket for x = 1.
Step 4: Number line representation of above intervals.
Step 5: Take random numbers between each interval and check if it satisfies the value. If it satisfies, then include interval in the solution.
For interval (-∞, 1) let random value be 0.
Putting x = 0 in the inequality (x + 3) / (x – 1) < 2
⇒ (0 + 3) / (0 – 1) < 2
⇒ 3 / (-1) < 2
⇒ -3 < 2 (True)
For interval (1, 5) let random value be 2.
Putting x = 3 in the inequality (x + 3) / (x – 1) < 2
⇒ (3 + 3) / (3 – 1) < 2
⇒ 6 / 2 < 2
⇒ 3 < 2 (False)
For interval (5, ∞) let random value be 2.
Putting y = 6 in the inequality (x + 3) / (x – 1) < 2
⇒ (6 + 3) / (6 – 1) < 2
⇒ 9 / 5 < 2
⇒ 1.8 < 2 (True)
Step 6: So, the solution for the absolute value inequality (x + 3) / (x – 1) < 2 is interval (-∞, 1) ∪ (5, ∞) as it satisfies the inequality which can be plotted on the number line as:
Inequalities
Inequalities are the expressions which define the relation between two values which are not equal. i.e., one side can be greater or smaller than the other. Inequalities are mathematical expressions in which both sides are not equal. They are used to compare two values or expressions. It is a mathematical expression used to compare the relative size or order of two objects or values.
They are fundamental in solving problems in mathematics, economics, engineering, and various other fields.
In this article, we will learn about Inequalities including their symbols, rules/properties, types, and their graphical representations and others in detail.
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