Inequalities are mathematical expressions that show the relationship between two values that are not necessarily equal. They use inequality symbols to compare quantities. The most common inequality symbols are:
For example, the inequality 3 < 5 means 3 is less than 5, while x ≥ 4 indicates that x is greater than or equal to 4. To solve the inequalities, we can use the following steps:
- Step 1: Write the inequality in the form of an equation.
- Step 2: Solve the equation and obtain the roots of the inequalities.
- Step 3: Represent the obtained values on the number line.
- Step 4: Represent the excluded values also on the number line with the open circles.
- Step 5: Find the intervals from the number line.
- Step 6: Take a random value from each interval, put these values in the inequality, and check if it satisfies the inequality.
- Step 7: The solution for the inequality is the intervals that satisfy the inequality.
Example: Solve the inequality x2 - 7x + 6 ≥ 0
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 = 1From 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 inequality x2 - 7x + 6 ≥ 0 is the interval (-∞, 1] ∪ [6, ∞) as it satisfies the inequality which can be plotted on the number line as:
Solving Polynomial Inequalities
Polynomial Inequalities include linear inequalities, quadratic inequalities, cubic inequalities, etc.
Solving Linear Inequalities
Linear inequalities can be solved like linear equations but according to the inequalities rule. Linear inequalities can be solved using simple algebraic operations.
One-Step Inequalities: One-step inequalities are inequalities that can be solved in one step.
Example: Solve: 5x < 10
⇒ 5x < 10 [Dividing both sides by 5]
⇒ x < 2 or (-∞, 2)
Two-Step Inequalities: Two-step inequalities are inequalities that can be solved in two steps.
Example: Solve: 4x + 2 ≥ 10
⇒ 4x + 2 ≥ 10
⇒ 4x ≥ 8 [Subtracting 2 from both sides]
⇒ 4x ≥ 8 [Dividing both sides by 4]
⇒ x ≥ 2 or [2, ∞)
Solving Rational Inequalities
A rational inequality is an inequality that contains a fraction with a variable in the numerator, denominator, or both.
Example: Solve the inequality: (x + 3) / (x - 1) < 2
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 inequality (x + 3) / (x - 1) < 2 is interval (-∞, 1) ∪ (5, ∞) as it satisfies the inequality which can be plotted on the number line as:
Solving Linear Inequality with Two Variables
Example: Solve: 20x + 10y ≤ 60
Consider x = 0 and put it in the given inequality
⇒ 20x + 10y ≤ 60
⇒ 20(0) + 10y ≤ 60
⇒ 10y ≤ 60
⇒ y ≤ 6 ------(i)
Now, when x = 0, y can be 0 to 6.
Similarly, putting values in inequality and check it satisfies the inequality.
For x = 1, y can be 0 to 4.
For x = 2, y can be 0 to 2.
For x = 3, y can be 0.
The possible solution for given inequality is (0, 0), (0,1), (0, 2), (0,3), (0,4), (0,5), (0,6), (1,0), (1,1), (1,2), (1,3), (1,4), (2,0), (2,1), (2,2), (3,0).
Solving Compound Inequalities
Compound inequalities are inequalities that have multiple inequalities separated by "and" or "or." To solve compound inequalities, solve the inequalities separately, and for the final solution, perform the intersection of obtained solutions if the inequalities are separated by "and" and perform the union of obtained solutions if the inequalities are separated by "or."
Example: Solve: 4x + 6 < 10 and 5x + 2 < 12
First solve 4x + 6 < 10
⇒ 4x + 6 < 10 [Subtracting 6 from both sides]
⇒ 4x < 4
⇒ x < 1 or (-∞, 1) ---(i)Second solve 5x + 2 < 12
⇒ 5x + 2 < 12 [Subtracting 2 from both sides]
⇒ 5x < 10
⇒ x < 2 or (-∞, 2) ---(ii)From (i) and (ii) we have two solutions x < 1 and x < 2. We take intersection for the final solution as the inequalities are separated by and.
⇒ (-∞, 1) ∩ (-∞, 2)
⇒ (-∞, 1)The final solution for given compound inequality is (-∞, 1).
Solving Absolute Value Inequalities
An absolute value inequality is an inequality that contains an absolute value expression. It shows how far a number is from zero and compares it using signs like <, >, ≤, ≥.
Example: Solve the inequality: |y + 1| ≤ 2
Following are the steps to solve inequality: |y + 1| ≤ 2
Step 1: Write the inequality in the form of an equation:
|y + 1| = 2
Step 2: Solve the equation:
y + 1 = ∓ 2
y + 1 = 2 and y + 1 = - 2
y = 1 and y = -3
From above step we obtain values y = 1 and y = -3
Step 3: From above values the intervals are (-∞, -3], [-3, 1], [1, ∞)
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 (-∞, -3] let random value be -4.
Putting y = -4 in the inequality |y + 1| ≤ 2
⇒ |-4+ 1| ≤ 2
⇒ |-3| ≤ 2
⇒ 3 ≤ 2 (False)
For interval [-3, 1] let random value be 0.
Putting y = 0 in the inequality |y + 1| ≤ 2
⇒ |0+ 1| ≤ 2
⇒ |1| ≤ 2
⇒ 1 ≤ 2 (True)
For interval [1, ∞) let random value be 2.
Putting y = 2 in the inequality |y + 1| ≤ 2
⇒ |2+ 1| ≤ 2
⇒ |3| ≤ 2
⇒ 3 ≤ 2 (False)
Step 6: So, the solution for the absolute value inequality |y + 1| ≤ 2 is interval [-3, 1] as it satisfies the inequality which can be plotted on the number line as:
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