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Definition
Formula Procedure Example 2x - 3 > 7 2x > 7 + 3 2x > 10 x > 10/ x > 5 The solution is x > 5. 2 Definition
Formula Procedure
Example 3x + 2 < 8 3x < 8 - 2 3x < 6 x < 6/ x < 2 The solution is x < 2.
Inequality An inequality is a mathematical expression that describes a relationship between two quantities, indicating that one is less than, greater than, or not equal to the other.
as explained. Solve linear inequalities using the same principles as solving linear equations.
as explained. Solve inequalities to find the possible values of the variable. Solve the inequality 2x - 3 > 7.
Linear Inequality A linear inequality is an inequality in which the highest power of the variable is 1.
Solve the linear inequality 3x + 2 < 8.
Definition • Formula • Procedure • Examples & Solutions
STUDY GUIDE
Mathematics: ALGEBRA
INEQUALITIES
Definition
Formula Procedure
Example
Definition
Formula Procedure
Example
Definition
Formula Procedure
Quadratic Inequality A quadratic inequality is an inequality in which the highest power of the variable is 2. as explained.
Compound Inequality A compound inequality is formed by combining two or more inequalities with the words "and" or "or." as explained. Solve compound inequalities by solving each individual inequality and considering the logical connection.
x < 4 Case 2: -(2x - 3) < 5 -2x + 3 < 5
Solve quadratic inequalities by factoring or analyzing the sign of the quadratic expression. Solve the quadratic inequality x^2 - 4x + 3 > 0.
Absolute Value Inequality An absolute value inequality involves the absolute value of a variable and typically includes expressions like |x|. as explained. Solve absolute value inequalities by considering both positive and negative cases.
x < 1 (Interval I)
Analyze the signs in the intervals:
(x - 3)(x - 1) > 0
Factor the quadratic expression:
Consider two cases: Case 1: 2x - 3 < 5 2x < 5 + 3 2x < 8
The solution is x < 1 or 1 < x < 3 or x > 3.
x > 3 (Interval III)
1 < x < 3 (Interval II)
Solve the absolute value inequality |2x - 3| < 5.
The solution is -1 < x < 4.
-2x < 5 - 3 -2x < 2 x > - Combine the solutions:
Procedure
Example
Definition
Formula Procedure
Example
Definition
Formula Procedure
Example
Definition
Formula
3y < 7 + 5 3y < 12 y < 12/ y < 4 The solution is y < 4.
Solve less than inequalities by isolating the variable on one side of the inequality sign. Solve the inequality 3y - 5 < 7. 3y - 5 < 7
Less Than or Equal To Inequality
2a + 3 ≥ 9 2a ≥ 9 - 3 2a ≥ 6 a ≥ 6/ a ≥ 3 The solution is a ≥ 3.
Greater Than or Equal To Inequality A greater than or equal to inequality (≥) expresses that one quantity is greater than or equal to another. as explained. Solve greater than or equal to inequalities by isolating the variable on one side of the inequality sign. Solve the inequality 2a + 3 ≥ 9.
A less than or equal to inequality (≤) expresses that one quantity is less than or equal to another. as explained. Solve less than or equal to inequalities by isolating the variable on one side of the inequality sign. Solve the inequality 5b - 2 ≤ 8.
Multiplication Property of Inequalities The multiplication property of inequalities states that if a > b and c is a positive number, then ac > bc. If c is a negative number, then ac < bc, but the inequality sign is reversed.
5b - 2 ≤ 8 5b ≤ 8 + 2 5b ≤ 10 b ≤ 10/ b ≤ 2 The solution is b ≤ 2.
as explained.
Procedure
Example
Definition
Formula Procedure
Example
Definition
Formula Procedure
Example Solve the inequality 3(x + 2) - 4 < 5. Start by distributing the 3: 3x + 6 - 4 < 5 Combine like terms: 3x + 2 < 5 Subtract 2 from both sides: 3x + 2 - 2 < 5 - 2 3x < 3
Solve multi-step inequalities by applying the properties of inequalities step by step.
2x - 3 > 7
The solution is x > 5. Solving Multi-Step Inequalities Multi-step inequalities require multiple steps to isolate the variable and find the solution. as explained.
x > 5
x > 10/
2x > 10
2x - 3 + 3 > 7 + 3
Addition Property of Inequalities The addition property of inequalities states that if a > b and c is a positive number, then a + c > b + c. If c is a negative number, then a + c < b + c, but the inequality sign is reversed. as explained. Apply the addition property to solve inequalities involving addition or subtraction.
Solve the inequality 2x - 3 > 7. Add 3 to both sides:
x > - The solution is x > -3.
Apply the multiplication property to solve inequalities involving multiplication or division. Solve the inequality -2x < 6. Divide both sides by -2, remembering to reverse the inequality sign: -2x < 6 x > 6/(-2)
Definition
Formula Procedure Example To find the union, consider the values that satisfy either of the inequalities: x < - x > 3 The values that satisfy either inequality are x < -2 or x > 3. So, the union is (-∞, -2) U (3, ∞). 17 Definition
Formula Procedure
Example Consider two cases: Case 1: 2x + 1 < 5 2x < 5 - 1 2x < 4 x < 4/ x < 2 Case 2: -(2x + 1) < 5 -2x - 1 < 5 -2x < 5 + 1 -2x < 6 x > 6/(-2) x > - Combine the solutions: -3 < x < 2 The solution is -3 < x < 2.
as explained.
Union of Inequalities The union of inequalities refers to finding values that satisfy at least one of the given inequalities.
Absolute Value Equations and Inequalities
Solve absolute value equations and inequalities by considering cases and analyzing the sign of the absolute value expression. Solve the absolute value inequality |2x + 1| < 5.
Absolute value equations and inequalities involve the absolute value of a variable and typically include expressions like |x|. as explained.
Find the union of solutions to two or more inequalities to determine the combined solution set.Find the values of x that satisfy either x < -2 or x > 3.
Definition
Formula Procedure
Example
Definition
Formula Procedure
Example For the first inequality: 2x + 3 > 1 2x > 1 - 3 2x > - x > -2/ x > - For the second inequality: x - 2 < 5 x < 5 + 2 x < 7 The intersection of solutions is -1 < x < 7.
as explained. Solve compound inequalities with "and" statements by finding the solutions to each inequality separately and taking their intersection. Solve the compound inequality 2x + 3 > 1 and x - 2 < 5.
2x < 7 - 3 2x < 4 x < 4/ x < 2
For the first inequality: 2x + 3 < 7
Compound Inequalities with "And" Statements Compound inequalities with "and" statements involve multiple inequalities connected by "and," indicating that the solution must satisfy all of the inequalities.
Compound Inequalities with "Or" Statements Compound inequalities with "or" statements involve multiple inequalities connected by "or," indicating that the solution can satisfy either of the inequalities. as explained. Solve compound inequalities with "or" statements by finding the solutions to each inequality separately and taking their union. Solve the compound inequality 2x + 3 < 7 or x - 2 > 4.
For the second inequality:
x > 6
x > 4 + 2
x - 2 > 4
The union of solutions is x < 2 or x > 6.
Definition
Formula Procedure
Example Consider two cases: Case 1: 2x + 1 < 5 2x < 5 - 1 2x < 4 x < 4/ x < 2 Case 2: -(2x + 1) < 5 -2x - 1 < 5 -2x < 5 + 1 -2x < 6 x > 6/(-2) x > - Combine the solutions: -3 < x < 2 The solution is -3 < x < 2.
23 Definition
Formula Procedure
Example
Solving inequalities with absolute values involves inequalities where absolute value expressions are present.
Solving Inequalities with Absolute Values
Solve the inequality |(3/2)x - 1| < 2/3.
Solving inequalities with absolute value fractions involves inequalities where absolute value expressions contain fractions.
Solving Inequalities with Absolute Value Fractions
as explained. Solve inequalities with absolute value fractions by considering cases and analyzing the sign of the absolute value expression while dealing with fraction arithmetic.
as explained. Solve inequalities with absolute values by considering cases and analyzing the sign of the absolute value expression. Solve the inequality |2x + 1| < 5.
(3/2)x < 5/
(3/2)x < 2/3 + 3/
(3/2)x < 2/3 + 1
Case 1: (3/2)x - 1 < 2/
Consider two cases:
x > 10/
(2/3) * (3/2)x > (5/3) * (2/3)
Multiply both sides by (2/3) (remembering to reverse the inequality sign for multiplication by a negative number):
Definition
Formula Procedure
Example Solve the inequality |x^2 - 4| < 3.
Solving inequalities with absolute value quadratics involves inequalities where the absolute value expressions contain quadratic expressions. as explained. Solve inequalities with absolute value quadratics by considering cases and analyzing the sign of the quadratic expression.
Solving Inequalities with Absolute Value Quadratics
-(3/2)x < 5/
-(3/2)x < 2/3 + 3/
-(3/2)x < 2/3 + 1
Multiply by -1 (remembering to reverse the inequality sign for multiplication by a negative number): x^2 > 1
The solution is -√7 < x < -1 or 1 < x < √7.
-√7 < x < √7 and x > 1 and x < -
Combine the solutions:
Take the square root (considering both positive and negative roots): x < √7 and x > -√ Case 2: -(x^2 - 4) < 3
-x^2 < -
-10/9 < x < 10/
Combine the solutions:
x > -10/
(2/3) * (-(3/2)x) > (5/3) * (2/3)
The solution is -10/9 < x < 10/9.
Case 2: -(3/2)x - 1 < 2/
Multiply both sides by (2/3) (remembering to reverse the inequality sign for multiplication by a negative number):
x^2 < 7
x^2 < 3 + 4
Case 1: x^2 - 4 < 3
Consider two cases:
x > 1 and x < -
Take the square root (considering both positive and negative roots):
-x^2 < 3 - 4
-x^2 + 4 < 3
Definition
Formula Procedure
Example
Take the square root (considering both positive and negative roots):
The solution is x > √7 or -1 < x < 1 or x < -√7.
Solve the absolute value inequality |x^2 - 4| > 3.
Solve absolute value inequalities with quadratics by considering cases and analyzing the sign of the quadratic expression.
Solving Absolute Value Inequalities with Quadratics Solving absolute value inequalities with quadratics involves inequalities where the absolute value expressions contain quadratic expressions. as explained.
-(3/2)x > 5/
The solution is -10/9 < x < 10/9.
-10/9 < x < 10/
Combine the solutions:
x > -10/
(2/3) * (-(3/2)x) > (5/3) * (2/3)
Multiply both sides by (2/3) (remembering to reverse the inequality sign for multiplication by a negative number):
-(3/2)x > 2/3 + 3/
-(3/2)x > 2/3 + 1
Case 2: -(3/2)x - 1 > 2/
Consider two cases:
-x^2 + 4 > 3
Case 2: -(x^2 - 4) > 3
x > √7 and x < -√
x^2 > 7
x^2 < 1
Multiply by -1 (remembering to reverse the inequality sign for multiplication by a negative number):
-x^2 > -
-x^2 > 3 - 4
x > √7 and x < -√7 or -1 < x < 1
Combine the solutions:
x < 1 and x > -
Take the square root (considering both positive and negative roots):
x^2 > 3 + 4
Case 1: x^2 - 4 > 3
Definition
Formula Procedure
Example Solve the absolute value inequality |(x - 2)/(x + 1)| > 1. Consider two cases: Case 1: (x - 2)/(x + 1) > 1
(x - 2) > 1(x + 1) x - 2 > x + 1 Subtract x from both sides: -2 > 1 This is not true, so there is no solution for this case. Case 2: -(x - 2)/(x + 1) > 1
-(x - 2) > 1(x + 1) -x + 2 > x + 1 Add x to both sides: 2 > 2x + 1 Subtract 1 from both sides: 1 > 2x
1/2 > x Combine the solutions: 1/2 > x The solution is x < 1/2. 29 Definition
Formula Procedure
as explained. Solve inequalities with absolute value and exponents by considering cases and analyzing the sign of the absolute value expression.
Solving Inequalities with Absolute Value and Exponents Solving inequalities with absolute value and exponents involves inequalities where the absolute value expressions contain exponential expressions.
Solving inequalities with absolute value and rational expressions involves inequalities where the absolute value expressions contain rational expressions.
Solving Inequalities with Absolute Value and Rational Expressions
as explained. Solve inequalities with absolute value and rational expressions by considering cases and analyzing the sign of the absolute value expression while dealing with rational expressions.
Multiply both sides by (x + 1) (remembering to reverse the inequality sign for multiplication by a negative number when x + 1 is negative):
Multiply both sides by (x + 1) (remembering to reverse the inequality sign for multiplication by a negative number when x + 1 is negative):
Divide by 2 (remembering to reverse the inequality sign for division by a negative number):
Definition
Formula Procedure
Example Solve the absolute value inequality |sin(x)| < 0.5. Consider two cases: Case 1: sin(x) < 0. Find the reference angle (inverse sine function): sin^(-1)(0.5) ≈ 30 degrees The solutions for x in this case are in the interval [-30, 30] degrees. Case 2: -(sin(x)) < 0.5 (and reverse the inequality sign) Find the reference angle (inverse sine function): sin^(-1)(0.5) ≈ 30 degrees
Combine the solutions: -30 < x < 30 degrees The solution is -30 < x < 30 degrees.
32 Definition
Formula Procedure
Example Solve the absolute value inequality |cos(x)| > 0.7.
as explained. Solve inequalities with absolute value and trigonometric functions by considering cases and analyzing the sign of the trigonometric expression.
Solving Absolute Value Inequalities with Trigonometric Expressions Solving absolute value inequalities with trigonometric expressions involves inequalities where the absolute value expressions contain trigonometric functions like sine, cosine, or tangent. as explained.
Solving Inequalities with Absolute Value and Trigonometric Functions Solving inequalities with absolute value and trigonometric functions involves inequalities where the absolute value expressions contain trigonometric expressions like sine, cosine, or tangent.
The solutions for x in this case are in the intervals [135, 225] degrees.
The solutions for x in this case are in the interval [-30, 30] degrees as well.
The solution is [0, 45] U [135, 225] U [315, 360] degrees.
[0, 45] U [315, 360] U [135, 225] degrees
Combine the solutions:
The solutions for x in this case are in the intervals [0, 45] degrees and [315, 360] degrees. Case 2: -(cos(x)) > 0.7 (and reverse the inequality sign) Find the reference angle (inverse cosine function): cos^(-1)(0.7) ≈ 45 degrees
Solve absolute value inequalities with trigonometric expressions by considering cases and analyzing the sign of the trigonometric expression.
Consider two cases: Case 1: cos(x) > 0. Find the reference angle (inverse cosine function): cos^(-1)(0.7) ≈ 45 degrees
