Solvw Quadratic Inequalities
Let’s take an example to solve absolute value inequalities.
Example: Solve the inequality: x2 – 7x + 6 ≥ 0
Solution:
Following are the steps to solve inequality: x2 – 7x + 6 ≥ 0
Step 1: Write the inequality in the form of equation:
x2 – 7x + 6 = 0
Step 2: Solve the equation:
x2 – 7x + 6 = 0
x2 – 6x – x + 6 = 0
x(x – 6) – 1(x – 6) = 0
(x – 6) (x – 1) = 0
x = 6 and x = 1
From above step we obtain values x = 6 and x = 1
Step 3: From above values the intervals are (-∞, 1], [1, 6], [6, ∞)
Since, the inequality is ≥ that includes equal to, so we use closed bracket for the obtained values.
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 -1.
Putting x = -1 in the inequality x2 – 7x + 6 ≥ 0
⇒ (-1)2 – 7(-1) + 6 ≥ 0
⇒ 1 + 7 + 6 ≥ 0
⇒ 14 ≥ 0 (True)
For interval [1, 6] let random value be 2.
Putting x = 0 in the inequality x2 – 7x + 6 ≥ 0
⇒ 22 – 7(2) + 6 ≥ 0
⇒ 4 – 14 + 6 ≥ 0
⇒ -4 ≥ 0 (False)
For interval [6, ∞) let random value be 7.
Putting x = 7 in the inequality x2 – 7x + 6 ≥ 0
⇒ 72 – 7(7) + 6 ≥ 0
⇒ 49 – 49 + 6 ≥ 0
⇒ 6 ≥ 0 (True)
Step 6: So, the solution for the absolute value inequality x2 – 7x + 6 ≥ 0 is the interval (-∞, 1] ∪ [6, ∞) 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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