5.3 Linear Inequalities in One Variable with Applications
4 min read•june 18, 2024
help us compare quantities and find ranges of solutions. They're like equations, but with greater-than or less-than symbols instead of equals signs. We use them to set limits or boundaries in math problems.
Solving inequalities involves similar steps to solving equations, but with a few twists. We can graph them on number lines, use , and apply them to real-world situations. Understanding inequalities is key to tackling many practical math problems.
Linear Inequalities in One Variable
Graphing linear inequalities
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- Inequalities compare two quantities using symbols
- represents "less than" (5 < 7)
- represents "greater than" (9 > 2)
- represents "less than or equal to" (x 4)
- represents "greater than or equal to" (y -1)
- Graphing inequalities on a involves
- Using an (○) for strict inequalities ( or )
- Using a (●) for inclusive inequalities ( or )
- Shading the portion of the number line that satisfies the inequality
- Shade to the right for or (x > 3 shades values greater than 3)
- Shade to the left for or (y ≤ 0 shades values less than or equal to 0)
- The shaded portion represents the of the inequality
- Interval notation expresses the of an inequality concisely
- Use parentheses ( ) for strict inequalities (x < 5 is written as (###-_0###, 5))
- Use brackets [ ] for inclusive inequalities (x ≥ -2 is written as [-2, ∞))
- represents negative infinity and represents positive infinity
- is written as in interval notation
Algebraic solutions for inequalities
- Solving linear inequalities algebraically is similar to solving linear equations
- Perform the same operation on both sides of the inequality to maintain the inequality (if x + 3 < 7, then x < 4)
- When multiplying or dividing by a negative number, reverse the inequality symbol
- If , then (dividing by -2 flips the inequality)
- Isolate the variable on one side of the inequality by
- Adding or subtracting the same value from both sides (x - 5 ≥ 1 becomes x ≥ 6)
- Multiplying or dividing both sides by the same positive value (2x < 10 becomes x < 5)
- Check your solution by substituting a value from the solution set into the original inequality (if x > 2, then 3 > 2 is true)
- The solution set represents the of the inequality
Linear Functions and Inequalities
- A is represented by an equation in the form y = mx + b
- m represents the of the line
- b represents the (where the line crosses the y-axis)
- Linear inequalities are closely related to linear functions
- They use the same symbols (>, <, ≥, ≤) to compare a linear expression to a constant or another linear expression
- A involves two or more inequalities that must be satisfied simultaneously
- A combines two inequalities using "and" or "or"
- For example, 2 < x < 5 means x is greater than 2 and less than 5
Real-world applications of inequalities
- Identify the unknown quantity and assign a variable (let x represent the number of tickets sold)
- Write an inequality that represents the or limits in the problem
- Translate verbal phrases into mathematical symbols
- "At least" means (the store needs to sell at least 50 items)
- "At most" means (the truck can carry at most 2,000 pounds)
- "More than" means (the company needs to produce more than 500 units)
- "Less than" means (the project must be completed in less than 30 days)
- Translate verbal phrases into mathematical symbols
- Solve the inequality algebraically (if 50 ≤ 2x + 10, then x ≥ 20)
- Interpret the solution in the context of the problem
- Check if the solution makes sense given the constraints (the company must produce at least 20 items to meet the minimum requirement)
- Example: A store requires a minimum purchase of $50 to qualify for free shipping. If sales tax is 6%, what is the minimum amount before tax needed to get free shipping?
- Let represent the amount spent before tax
- The total cost with tax must be at least 1.06x \geq 50$
- Solve the inequality:
- Interpret the result: You need to spend at least $47.17 before tax to qualify for free shipping
Key Terms to Review (31)
-∞: -∞, or negative infinity, represents a concept in mathematics indicating a value that is less than all real numbers. It is commonly used to describe limits, solutions to inequalities, and boundaries in mathematical contexts, particularly when analyzing behavior of functions and graphs. This notion allows for the exploration of ranges and solutions that extend infinitely in the negative direction.
<: The symbol '<' represents a mathematical inequality indicating that the value on the left is less than the value on the right. This symbol is used to compare two quantities and helps in formulating linear inequalities, which are essential in both real-world applications and graphical representations of relationships between variables.
>: The symbol '>' represents a greater than relation in mathematics, indicating that one value is larger than another. This concept is fundamental when dealing with inequalities, allowing us to express and analyze relationships between numbers or expressions. Understanding this symbol is essential for solving linear inequalities and systems of inequalities, as well as for evaluating fairness in voting methods where comparisons are necessary.
∞: The symbol ∞ represents infinity, a concept that describes something without any limit or bound. In mathematics, it is often used to signify the idea of quantities that can grow indefinitely or processes that do not have an endpoint. This concept is crucial in various mathematical contexts, as it helps in understanding limits, series, and behaviors of functions.
≤: The symbol '≤' represents a relationship in mathematics indicating that one value is less than or equal to another. This symbol is crucial for expressing inequalities and understanding the boundaries of solutions in various mathematical contexts, helping to determine feasible solutions, especially in optimization problems.
≥: The symbol '≥' represents the concept of 'greater than or equal to' in mathematics, establishing a relationship between two values. This symbol is crucial in expressing linear inequalities, where one side of the inequality can either exceed or be equal to the other side. It helps define boundaries in mathematical expressions and is foundational in various applications like optimization and systems of inequalities.
Addition Property: The Addition Property states that if you add the same value to both sides of an equation or inequality, the two sides remain equal or maintain their relationship. This principle is crucial in solving linear inequalities in one variable, as it allows for the manipulation of the inequality while ensuring that the solutions remain valid and equivalent.
Closed Circle: A closed circle is a graphical representation used in the context of inequalities to indicate that a particular value is included in the solution set. This is crucial when expressing inequalities, particularly when dealing with less than or equal to ($$\leq$$) and greater than or equal to ($$\geq$$) conditions. The closed circle visually communicates that the endpoint is part of the solution, highlighting key distinctions between inclusive and exclusive scenarios.
Compound inequality: A compound inequality is a mathematical statement that combines two or more inequalities using the words 'and' or 'or'. This concept allows for expressing a range of values that satisfy multiple conditions at once, making it particularly useful when analyzing relationships in linear inequalities and their applications. Understanding compound inequalities helps in solving real-world problems by determining feasible solutions within specified limits.
Constraint: A constraint is a limitation or restriction that defines the boundaries within which a solution must be found. In mathematical contexts, constraints help to narrow down possible solutions by establishing conditions that must be satisfied, such as inequalities or specific values. They are crucial for solving problems, as they guide the decision-making process in various applications.
Constraints: Constraints are conditions or limits imposed on variables in a mathematical model. They restrict the feasible region within which an optimal solution can be found.
Domain: The domain of a function is the set of all possible input values (typically represented as 'x') that the function can accept. It determines the range of values for which the function is defined and produces real numbers.
Domain: In mathematics, the domain refers to the complete set of possible values that a variable can take. It plays a crucial role in defining functions, inequalities, and equations as it establishes the input values for which these mathematical expressions are valid. Understanding the domain helps in determining where a function or equation can be applied and helps avoid undefined situations.
Feasible region: The feasible region is the set of all possible solutions that satisfy a given set of constraints in a mathematical context. This concept is crucial when dealing with inequalities and optimization problems, as it visually represents the area where all constraints overlap. The feasible region is often graphed in two-dimensional space, showing the combinations of variables that meet all specified conditions.
Interval Notation: Interval notation is a mathematical notation used to represent a range of numbers between two endpoints. This system allows for the concise expression of intervals, where the type of brackets indicates whether the endpoints are included or excluded, making it essential for understanding solutions to linear inequalities.
Linear function: A linear function is a mathematical relationship that produces a straight line when graphed, characterized by the equation of the form $$y = mx + b$$, where $$m$$ represents the slope and $$b$$ represents the y-intercept. This type of function maintains a constant rate of change, meaning that for every unit increase in the independent variable, there is a proportional change in the dependent variable. Linear functions can be used to model various real-world situations, allowing for predictions and analysis.
Linear inequalities: Linear inequalities are algebraic expressions involving a linear function where the values are not equal but rather greater than or less than each other. They can be represented on a number line or through interval notation.
Linear Inequality: A linear inequality is a mathematical expression that shows the relationship between two values where one value is not equal to the other, typically expressed in the form of $ax + b < c$, $ax + b \leq c$, $ax + b > c$, or $ax + b \geq c$. These inequalities can represent ranges of values rather than single points, and they are used to model situations where constraints exist, making them essential in understanding how to evaluate and compare different scenarios.
Multiplication Property: The multiplication property states that if you multiply both sides of an inequality by a positive number, the direction of the inequality remains unchanged. Conversely, if you multiply both sides by a negative number, the inequality sign flips. This property is crucial for solving linear inequalities and ensuring that the solutions maintain their validity within the context of mathematical operations.
Non-Strict Inequality: Non-strict inequality refers to relationships between quantities that allow for equality, represented by the symbols '≥' (greater than or equal to) and '≤' (less than or equal to). This concept is essential for understanding how ranges of values can satisfy an inequality, as it includes the boundary points themselves. Non-strict inequalities are particularly useful when modeling real-world situations, where a value can be equal to a certain threshold or limit.
Number line: A number line is a visual representation of numbers arranged in a straight horizontal line, where each point corresponds to a real number. It serves as a tool to understand numerical relationships and operations, providing a clear way to visualize integers, rational numbers, and real numbers. The concept of the number line is crucial for solving equations and inequalities, helping to illustrate how different types of numbers are related and how they can be compared or combined.
Open Circle: An open circle is a graphical representation used on a number line to indicate that a particular value is not included in the solution set of an inequality. It signifies that the endpoint of the interval is excluded, contrasting with a closed circle, which indicates inclusion. Open circles are essential for visualizing linear inequalities, particularly when solving equations or inequalities involving greater than (>) or less than (<) relationships.
Optimization: Optimization is the process of finding the best solution or outcome from a set of possible choices, often subject to certain constraints. It plays a vital role in decision-making where the goal is to maximize or minimize a specific function, whether it be cost, time, efficiency, or resources. This concept is applied in various fields to analyze and improve systems, ensuring that limited resources are used effectively.
Range: Range refers to the set of all possible output values (or dependent variable values) of a function, determined by the inputs in the domain. Understanding range is crucial as it helps to identify the limits of a function's output and how it behaves under different conditions, which can be connected to various mathematical concepts including inequalities, quadratic equations, and statistical measures.
Slope: Slope is a measure of the steepness or incline of a line, typically represented as the ratio of the vertical change to the horizontal change between two points on that line. It plays a crucial role in understanding relationships in equations and inequalities, helping to determine whether they increase or decrease, and is essential for graphing functions and analyzing systems of equations.
Solution set: A solution set is the collection of all values that satisfy a given equation or inequality. In mathematical contexts, this term is crucial as it helps identify all possible answers that make an equation true or meet the conditions of an inequality, creating a clearer understanding of relationships between variables.
Strict inequality: Strict inequality refers to a relationship between two values where one value is definitively less than or greater than another value, denoted by the symbols '<' for 'less than' and '>' for 'greater than'. Unlike regular inequalities that can include equality (like '≤' or '≥'), strict inequalities exclude the possibility of the two values being equal, creating a clear distinction between them. Understanding strict inequality is crucial in mathematical contexts where precise conditions are required, such as solving linear inequalities and modeling real-world applications.
System of inequalities: A system of inequalities is a collection of two or more inequalities that involve the same variables. It is used to describe a range of possible solutions that satisfy all the inequalities simultaneously, which can be represented graphically in a coordinate plane. The intersections and boundaries created by these inequalities are crucial for determining feasible regions, especially in real-world applications like optimization problems.
Transitive Property: The transitive property states that if one quantity is equal to a second quantity, and that second quantity is equal to a third quantity, then the first quantity is also equal to the third. This property allows for the comparison and manipulation of inequalities, which is essential when solving linear inequalities in one variable. Understanding this property helps in establishing relationships between various expressions and inequalities, making it easier to find solutions and draw conclusions.
Trichotomy Property: The trichotomy property states that for any two real numbers, one and only one of the following statements is true: the first number is less than the second, the first number is equal to the second, or the first number is greater than the second. This fundamental property is crucial in understanding linear inequalities, as it establishes a clear comparison between values, which is essential for solving inequalities and making decisions based on their results.
Y-intercept: The y-intercept is the point where a graph intersects the y-axis, representing the value of the dependent variable when the independent variable is zero. This key feature helps to understand linear relationships, curves, and data trends, providing crucial information for graphing and analyzing equations across various mathematical contexts.