Simplified Quantitative Formulas: Inequalities
- Inequality: A statement that two values are not equal, using symbols: <, >, ≤, ≥, ≠.
- Linear Inequality: ax + b < c, ax + b > c, ax + b ≤ c, ax + b ≥ c.
- Quadratic Inequality: ax² + bx + c < 0, > 0, ≤ 0, ≥ 0.
- Solving: Manipulate like equations, but reverse the sign when multiplying/dividing by a negative.
- Interval Notation: (a, b), [a, b], (a, b], [a, b).
- System of Inequalities: Solve each separately, then find the intersection.
- Key Properties: If a < b and b < c, then a < c. If a < b, then a + c < b + c. If a < b and c > 0, then ac < bc. If c < 0, ac > bc.
- AM ≥ GM ≥ HM: For positive numbers, Arithmetic Mean ≥ Geometric Mean ≥ Harmonic Mean.
- Variable Definitions: a, b, c = numbers; x = variable; AM = arithmetic mean; GM = geometric mean; HM = harmonic mean.
What do these mean? (Super Simple Explanations & Examples)
- Linear Inequality: 2x + 3 < 7 ⇒ 2x < 4 ⇒ x < 2.
- Quadratic Inequality: x² – 5x + 6 > 0 ⇒ (x–2)(x–3) > 0 ⇒ x < 2 or x > 3.
- Interval Notation: x ∈ (–∞, 2) ∪ (3, ∞).
- Reverse Sign: –2x < 4 ⇒ x > –2 (sign reverses when dividing by negative).
- System of Inequalities: x > 1 and x < 5 ⇒ x ∈ (1, 5).
- AM ≥ GM ≥ HM: For 2, 8: AM = 5, GM = 4, HM = 3.2 ⇒ 5 ≥ 4 ≥ 3.2.
- Variable Definitions: a, b, c = numbers; x = variable; AM = arithmetic mean; GM = geometric mean; HM = harmonic mean.
Basic Inequalities
Inequalities are mathematical expressions that compare two quantities using symbols like <, >, ≤, or ≥.
Basic Inequality Symbols
Inequalities use specific symbols to represent different types of relationships between numbers:
< : Less than
> : Greater than
≤ : Less than or equal to
≥ : Greater than or equal to
Example: 5 < 7 (5 is less than 7)
3 > 2 (3 is greater than 2)
4 ≤ 4 (4 is less than or equal to 4)
6 ≥ 5 (6 is greater than or equal to 5)
Example 1: Basic Inequality Operations
Q1.
If a > b and b > c, what can we say about a and c?
Solution:
By the transitive property of inequalities:
If a > b and b > c, then a > c
This is because a is greater than b, and b is greater than c, so a must be greater than c
Q2.
If x < y, what happens to the inequality when we multiply both sides by -2?
Solution:
When multiplying or dividing both sides of an inequality by a negative number, the inequality sign reverses
x < y
-2x > -2y (sign reversed)
Q3.
If a ≥ b and c > 0, what can we say about ac and bc?
Solution:
When multiplying both sides of an inequality by a positive number, the inequality sign remains the same
a ≥ b and c > 0
Therefore, ac ≥ bc
Q4.
If x ≤ y and y ≤ z, what can we say about x and z?
Solution:
By the transitive property of inequalities:
If x ≤ y and y ≤ z, then x ≤ z
This is because x is less than or equal to y, and y is less than or equal to z, so x must be less than or equal to z
Linear Inequalities
Linear inequalities are inequalities that can be written in the form ax + b < 0, ax + b > 0, ax + b ≤ 0, or ax + b ≥ 0.
Linear Inequality Forms
Linear inequalities can be written in various forms, and they can be solved using similar methods as linear equations, with special attention to the direction of the inequality sign.
General forms:
ax + b < 0
ax + b > 0
ax + b ≤ 0
ax + b ≥ 0
Example: 2x + 3 > 7
2x > 7 - 3
2x > 4
x > 2
Example 1: Solving Linear Inequalities
Q1.
Solve the inequality: 3x + 5 > 14
Solution:
3x + 5 > 14
3x > 14 - 5
3x > 9
x > 3
Q2.
Solve the inequality: -2x + 4 ≤ 10
Solution:
-2x + 4 ≤ 10
-2x ≤ 10 - 4
-2x ≤ 6
x ≥ -3 (sign reversed when dividing by negative)
Q3.
Solve the inequality: 4(x - 2) ≥ 2x + 6
Solution:
4(x - 2) ≥ 2x + 6
4x - 8 ≥ 2x + 6
4x - 2x ≥ 6 + 8
2x ≥ 14
x ≥ 7
Q4.
Solve the inequality: (x + 3)/2 < (2x - 1)/3
Solution:
(x + 3)/2 < (2x - 1)/3
3(x + 3) < 2(2x - 1)
3x + 9 < 4x - 2
9 + 2 < 4x - 3x
11 < x
x > 11
Quadratic Inequalities
Quadratic inequalities are inequalities that can be written in the form ax² + bx + c < 0, ax² + bx + c > 0, ax² + bx + c ≤ 0, or ax² + bx + c ≥ 0.
Solving Quadratic Inequalities
To solve quadratic inequalities, we first find the roots of the corresponding quadratic equation, then use these roots to determine the intervals where the inequality holds true.
Steps:
1. Find the roots of ax² + bx + c = 0
2. Plot the roots on a number line
3. Test a value in each interval
4. Determine the solution set
Example: x² - 4x + 3 > 0
Roots: x = 1, 3
Test x = 0: 3 > 0 (true)
Test x = 2: -1 > 0 (false)
Test x = 4: 3 > 0 (true)
Solution: x < 1 or x > 3
Example 1: Solving Quadratic Inequalities
Q1.
Solve the inequality: x² - 5x + 6 > 0
Solution:
First, find the roots of x² - 5x + 6 = 0
(x - 2)(x - 3) = 0
x = 2 or x = 3
Test x = 0: 6 > 0 (true)
Test x = 2.5: -0.25 > 0 (false)
Test x = 4: 2 > 0 (true)
Solution: x < 2 or x > 3
Q2.
Solve the inequality: x² + 2x - 3 ≤ 0
Solution:
First, find the roots of x² + 2x - 3 = 0
(x + 3)(x - 1) = 0
x = -3 or x = 1
Test x = -4: 5 ≤ 0 (false)
Test x = 0: -3 ≤ 0 (true)
Test x = 2: 5 ≤ 0 (false)
Solution: -3 ≤ x ≤ 1
Q3.
Solve the inequality: 2x² - 7x + 3 < 0
Solution:
First, find the roots of 2x² - 7x + 3 = 0
Using quadratic formula: x = (7 ± √(49 - 24))/4
x = (7 ± 5)/4
x = 3 or x = 0.5
Test x = 0: 3 < 0 (false)
Test x = 1: -2 < 0 (true)
Test x = 4: 7 < 0 (false)
Solution: 0.5 < x < 3
Q4.
Solve the inequality: x² - 4x + 4 ≥ 0
Solution:
First, find the roots of x² - 4x + 4 = 0
(x - 2)² = 0
x = 2 (double root)
Test x = 0: 4 ≥ 0 (true)
Test x = 2: 0 ≥ 0 (true)
Test x = 4: 4 ≥ 0 (true)
Solution: All real numbers
Absolute Value Inequalities
Absolute value inequalities involve expressions with absolute value signs and can be solved by considering different cases.
Absolute Value Inequality Rules
Absolute value inequalities can be solved using specific rules based on the inequality sign and the value being compared.
For |x| < a: -a < x < a
For |x| > a: x < -a or x > a
For |x| ≤ a: -a ≤ x ≤ a
For |x| ≥ a: x ≤ -a or x ≥ a
Example: |2x - 3| < 5
-5 < 2x - 3 < 5
-2 < 2x < 8
-1 < x < 4
Example 1: Solving Absolute Value Inequalities
Q1.
Solve the inequality: |x - 2| < 3
Solution:
|x - 2| < 3
-3 < x - 2 < 3
-3 + 2 < x < 3 + 2
-1 < x < 5
Q2.
Solve the inequality: |2x + 1| ≥ 5
Solution:
|2x + 1| ≥ 5
2x + 1 ≤ -5 or 2x + 1 ≥ 5
2x ≤ -6 or 2x ≥ 4
x ≤ -3 or x ≥ 2
Q3.
Solve the inequality: |3x - 2| ≤ 4
Solution:
|3x - 2| ≤ 4
-4 ≤ 3x - 2 ≤ 4
-4 + 2 ≤ 3x ≤ 4 + 2
-2 ≤ 3x ≤ 6
-2/3 ≤ x ≤ 2
Q4.
Solve the inequality: |x + 3| > 2
Solution:
|x + 3| > 2
x + 3 < -2 or x + 3 > 2
x < -5 or x > -1
Rational Inequalities
Rational inequalities involve fractions with variables in the denominator and require special attention to the domain.
Solving Rational Inequalities
To solve rational inequalities, we need to find the critical points (zeros and undefined points) and test intervals between them.
Steps:
1. Find values that make numerator or denominator zero
2. Plot these values on a number line
3. Test a value in each interval
4. Consider the domain restrictions
Example: (x - 2)/(x + 1) > 0
Critical points: x = 2, x = -1
Test x = -2: 4 > 0 (true)
Test x = 0: -2 > 0 (false)
Test x = 3: 1/4 > 0 (true)
Solution: x < -1 or x > 2
Example 1: Solving Rational Inequalities
Q1.
Solve the inequality: (x - 3)/(x + 2) > 0
Solution:
Critical points: x = 3, x = -2
Test x = -3: 6 > 0 (true)
Test x = 0: -1.5 > 0 (false)
Test x = 4: 1/6 > 0 (true)
Solution: x < -2 or x > 3
Q2.
Solve the inequality: (2x + 1)/(x - 1) ≤ 0
Solution:
Critical points: x = -0.5, x = 1
Test x = -1: 1/2 ≤ 0 (false)
Test x = 0: -1 ≤ 0 (true)
Test x = 2: 5 ≤ 0 (false)
Solution: -0.5 ≤ x < 1
Q3.
Solve the inequality: (x² - 4)/(x + 3) ≥ 0
Solution:
Critical points: x = -2, x = 2, x = -3
Test x = -4: 12 ≥ 0 (true)
Test x = 0: -4/3 ≥ 0 (false)
Test x = 3: 5/6 ≥ 0 (true)
Solution: x < -3 or -2 ≤ x ≤ 2
Q4.
Solve the inequality: (x + 1)/(x - 2) < 0
Solution:
Critical points: x = -1, x = 2
Test x = -2: 1/4 < 0 (false)
Test x = 0: -0.5 < 0 (true)
Test x = 3: 4 < 0 (false)
Solution: -1 < x < 2
Advanced Concepts
Advanced topics in inequalities including compound inequalities, systems of inequalities, and applications.
Advanced Inequality Concepts
Advanced inequality concepts include compound inequalities, systems of inequalities, and their applications in various fields.
Compound inequalities: a < x < b
Systems of inequalities: Multiple inequalities that must be satisfied simultaneously
Applications: Optimization problems, constraints in real-world scenarios
Example: Solve the system:
2x + y > 4
x - y < 1
Solution: Graph the inequalities and find the intersection region
Example 1: Compound Inequalities
Q1.
Solve the compound inequality: -3 < 2x + 1 < 5
Solution:
-3 < 2x + 1 < 5
-3 - 1 < 2x < 5 - 1
-4 < 2x < 4
-2 < x < 2
Q2.
Solve the system of inequalities: x + y > 3 and 2x - y < 4
Solution:
First inequality: y > 3 - x
Second inequality: y > 2x - 4
The solution is the region where both inequalities are satisfied
This can be found by graphing both inequalities and finding their intersection
Q3.
Solve the compound inequality: 1 ≤ 3x - 2 ≤ 7
Solution:
1 ≤ 3x - 2 ≤ 7
1 + 2 ≤ 3x ≤ 7 + 2
3 ≤ 3x ≤ 9
1 ≤ x ≤ 3
Q4.
Solve the system of inequalities: x > 0, y > 0, and x + y < 6
Solution:
This system defines a region in the first quadrant
x > 0 and y > 0: First quadrant
x + y < 6: Below the line x + y = 6
The solution is the triangular region bounded by the axes and the line x + y = 6
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Practice Inequalities Questions