Mathematics
Equations and Inequalities
Equations and inequalities are fundamental concepts in mathematics that involve the use of symbols and mathematical operations to express relationships between quantities. Equations are statements of equality, while inequalities express relationships where one quantity is greater than, less than, or not equal to another. Solving equations and inequalities often involves finding the values of variables that satisfy the given conditions.
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eBook - ePub- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
extraneous solutions , solutions that are incorrect. So we need to check our solutions in the original equation.EXAMPLE
Solve for x : .SOLUTION
5x − 6 = 95x = 15x = 3Check: .EXAMPLE
Solve for x : .SOLUTION
2x + − = 162x = 14x = 7.Check: .Therefore, this equation has no solution.Inequalities
An inequality is a statement that the value of one quantity or expression is greater than or less than that of another. There are five inequality symbols:> greater than < less than ≥ greater than or equal to ≤ less than or equal to ≠ not equal toWhen we have inequalities with variables, there are certain values of the variable that may make the inequality true and others that make it false. We can show solutions to inequalities by using a number line. An open dot (°) at a value means the inequality does not include that value, and a closed dot (•) means that it does includes that value.The rules for solving simple linear inequalities are the same as those for linear equations with one exception. If we multiply or divide both sides of an inequality by a negative number, the direction of the inequality changes. To check a solution, plug a number within the solution into the original inequality to see whether it is true. To be extra sure, you can choose a number not in the solution interval and be sure that it does not
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
10SOLVING INEQUALITIES
WHAT YOU WILL LEARN
•How to write an inequality to represent a constraint or condition in a real-world or mathematical problem •How to recognize that inequalities have infinitely many solutions, and represent solutions of such inequalities on number line diagrams •How to solve inequalities and represent the solution set •How to write inequalities to solve problems by reasoning about the quantities •How to solve word problems leading to inequalities •How to graph the solution set of an inequality and interpret it in the context of the problemSECTIONS IN THIS CHAPTER•What Is an Inequality?•How Do We Represent Solutions of Inequalities?•How Do We Solve Inequalities?•How Can We Use Inequalities to Solve Word Problems?10.1 What Is an Inequality?
DEFINITION
InequalityA mathematical statement containing one of the symbols >, <, ≥, ≤, or ≠ to indicate the relationship between two quantities.An inequality tells you when things are not equal. There are many times when an exact number isn’t needed. Think about situations when you are given a minimum or a maximum. Those situations generate inequalities.EXAMPLE:A curfew of 11 P.M . means be home at or before 11 P.M .You can come home at 9 P.M ., 10 P.M ., or 10:30 P.M .—even 11 P.M .You’d better not come home at 11:30; you would be in trouble. You have to be at least 18 years old to vote. You can be 18, 19, 25, or even 75 (like my Uncle Carl). You can’t be 16, 12, or even 4 (like Charlie). You can be at most 12 years old to order from the kid’s menu. You can be 12, 11, or even just a few months (like Luke). You can’t be 13, 16, or 26 (like Chris). A party room has a maximum capacity of 125 people. You can have 125, 124, or only 6. You can’t have 126, 130, or 250. The fire marshal will shut it down.
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
An equivalent inequality results when the same quantity is added or subtracted on each side of the inequality. For example, since 4 < 8, it is also true that•Multiplication and division property of inequalitiesAn equivalent inequality results when each side of an inequality is multiplied or divided by the same positive quantity. For example, since 4 < 8, it is also true thatAn equivalent inequality results when each side of an inequality is multiplied or divided by the same negative quantity and the direction of the inequality sign is reversed. For example, if −2x < 6, thenSOLVING A LINEAR INEQUALITY
You can solve a linear inequality as you would an equation except that, when both sides of an inequality are multiplied or divided by the same negative number, you must reverse the sign of the inequality. To find the solution set of 1 – 2x ≤ x + 13:SET-BUILDER NOTATION
A set is a collection of objects. A set is written in roster form when its members are listed individually within braces, as in A = {7, 8, 9, 10, 11}. Set-builder notation replaces the individual elements in a set with a general rule for determining whether or not a particular number is a member of the set. Using set-builder notation:Set A is read as “the set of all x such that x is greater than or equal to 7 and less than or equal to 11, where x is a positive integer.” Sometimes a colon is used instead of a vertical bar, as in A = {x : 7 ≤ x ≤ 11; x is a positive integer}. The colon and vertical bar are each translated as “such that.” If the replacement set for x is not indicated within the braces, assume it is the largest possible subset of real numbers. For example, {x | x > 3} represents the set of all real numbers greater than 3.INTERVAL NOTATION
An interval
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
a can be any number less than 7, such as –3, 0, 5, 6, but not 7 or anything greater than 7.2. a > b a “is greater than” b. So if a > 4, a can be 5, 7, 100, or any number more than 4, but not 4 or anything less than 4.3. a ≤ b a “is less than or equal to” b. So if a ≤ 3 1 7, then a is any number less than 10, including 10.4. a ≥ b a “is greater than or equal to” b. So if a ≥ 6 – 2, then a is any number greater than 4, including 4.5. a ≠ b a and b aren’t equal (it doesn’t say which is larger). So if a is 5, then b cannot be 5.Solving Linear Inequalities
Inequalities can be combined to indicate a specific range of values. For example, – 2 ≤ x < 6, with x being a whole number, means x can be –2, –1, 0, 1, 2, 3, 4, or 5. A set of numbers is sometimes written with braces, such as {–2, –1, 0, 1, 2, 3, 4, 5}. Notice that it includes –2 here but not 6.Working with inequalities is similar to working with equalities—whatever we do to one side, we must also do to the other side of the inequality. The only difference is that if we multiply or divide an inequality by a negative, the inequality sign switches. This is due to the hierarchy of negative numbers: although 5 is less than 7, –5 is greater than –7.Inequalities are used if there is a limit. For example, if you have only $10 to spend at a store, your purchases (plus tax) must be ≤ $10.If we multiply or divide an inequality by a negative, the inequality sign switches. To help you remember this, just think of the number line, where –10 < –5 < but 10 > 5. If we multiply (or divide) –10 < –5 by –1 < we get 10 > 5. The inequality sign switches.
Jamal is on commission and he wants to earn at least $1,000 in a certain amount of time. His commission is $40 per sale. How many sales does he have to make to meet his goal?Example 5.23.Answer 5.23.The inequality to use is sales × commission ≥ $1,000. In this case, let’s let sales = x, and the inequality becomes 40x ≥ 1000. Dividing by 40 on both sides of the inequality, x
eBook - ePubGMAT Advanced Quant
250+ Practice Problems & Online Resources
- (Author)
- 2020(Publication Date)
- Manhattan Prep(Publisher)
- Some Word Problems can create a hidden constraint involving inequalities. These inequalities may come into play in determining the correct answer. For example, a problem might read: “The oldest student in the class . . . the next oldest student in the class . . . the youngest student in the class . . .” This can be translated to the following inequality: youngest < middle < oldest.
- Inequalities involving a variable in a denominator often involve two possibilities: a positive and a negative one. For example, if you know that , you might be tempted to multiply by y and arrive at 1 < xy. However, this may not be correct. It depends on whether y is a positive or negative number. If y > 0, then it is correct to infer that 1 < xy. However, if y < 0, then 1 > xy. Therefore, you’ll need to test two cases (positive and negative) in this situation.
- At the same time, hidden constraints may allow you to manipulate inequalities more easily. For instance, if a quantity must be positive, then you can multiply both sides of an inequality by that quantity without having to set up two cases.
- Many questions involving inequalities are actually disguised positive/negative questions. For example, if you know that xy > 0, the fact that xy is greater than 0 is not in and of itself very interesting. What is interesting is that the product is positive, meaning both x and y are positive or both x and y are negative. Thus, x and y have the same sign. Here, the inequality symbol is used to disguise the fact that x and y have the same sign.
Take a look at some examples that illustrate these concepts.Try-It #4-4
If is a prime number, what is the value of x ?- −16 < −3x + 5<22
- x2 is a two-digit number
If is a prime number, then possible values are 2, 3, 5, and so on. Therefore, x must be a perfect square of a prime; possible values include 4, 9, 25, and so on.(1) SUFFICIENT: Manipulate the inequality to isolate x:Since x is the square of a prime, it can’t be negative or zero; it has to be positive. The smallest possible square of a prime is 4 and the next smallest possible square of a prime is 9. This inequality allows just one possible value: x
eBook - ePubA Cultural History of Reforming Math for All
The Paradox of Making In/equality
- Jennifer D. Diaz(Author)
- 2017(Publication Date)
- Routledge(Publisher)
In school mathematics the equal sign rarely appears alone. It makes sense in relation to other symbols and objects. The equal sign and numbers organize problems for children to solve about identity and difference from which to understand the fact of equality. When children are asked to determine this equality, they are to see and think about the numerical statements of equivalence as a representation of some truth in the world.Are these math sentences true (T) or false (F)? Why?4 + 5 = 9 7 = 3 + 4 8 = 8 10–5 = 2 + 3 8 + 2 = 10 + 4 (Beatty & Moss, 2007, p. 31)The numbers and the equal sign appear to establish equivalences and difference, giving meaning to the expressions as either true or false. Here the numbers do not make sense in and of themselves. They are validated as social facts and involved in constructing the social world, not just representing it (Porter, 1995). In school math problems, like those here, the numbers work as a strategy for constructing a truth about equivalence that is seen as a mathematical form of reasoning. This highlights how the use of “numbers in the representation of the world is predicated upon procedures of classification and separation of the ‘identical’ and ‘different’ which results in the building of a rigid perception of reality” (Patriarca, 1996, p. 9). More than mathematics, the equation also embodies rules for how to see difference, sameness, equalities, and inequalities as a form of knowing.The practice of classifying statements by their truth value embodies a way of categorizing relationships in the world. The problem-solving here can be considered to inscribe in the child a “style of scientific reasoning” (Hacking, 2002). Carrying a way of thinking, the expressions of in/equality carry rules that define equality as a relationship of sameness. These rules then are to be internalized and applied in a presumably rational way to other instantiations of equality represented with the equal sign.
eBook - ePub- Daniel Greenberg(Author)
- 2014(Publication Date)
- Research & Education Association(Publisher)
5Algebra Topics
5.1 VARIABLES AND EXPRESSIONSAn expression is a mathematical phrase that contains numbers, variables, and symbols such as +, −, ×, and ÷. An expression can contain parentheses or other symbols as well. Numbers and variables that are separated by + or − signs are called terms ; those that are part of multiplication are called factors .An example of a simple algebraic expression is just the variable x . An expression may contain two terms, such as 2a + 3b , or it can be quite complicated, involving many letters, numbers, and symbols.A variable is a term that stands for a number or quantity, but its value can change (in contrast to a letter such as e , the elementary electron charge, which is a constant). Usual letters used for variables in algebra are x , y , z , a , b , c , but they can be anything.Multiplication can be indicated by x, •, or *, or by parentheses, as in (2)(3x ), which equals 6x , or even by nothing, as in xy , which means multiply x by y .A. Simplifying Expressions
Simplifying an expression is presenting it in a simpler form, perhaps by removing parentheses or combining some terms.5.1Problem 1: Which expression is greater: 5 − (8) or −(28)?A) 5 − (8) is greater.B) 5(−8) is greater.C) The expressions are equal in value.D) Both expressions equal −3.STRATEGY
Remove parentheses; then simplify.THINK
• Write the expression. Insert signs to give every term a sign.• Resolve signs using simple rules.• Remove parentheses and simplify. −3 is greater than −40, so 5 − (8) is greater. The correct answer choice is (A).B. Properties
The basic properties of expressions are called the associative, commutative, and distributive properties. These properties allow us to simplify expressions.• Commutative
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
7 Solution of Algebraic EquationsSince equations reflect the major physicochemical principles that govern equipment operation, their solution conveys specific values for parameters that were not arbitrated (or somehow fixed in advance). A major distinction is, however, be put forward between exact analytical solutions – either of closed or open forms; the former encompasses linear equations and the quadratic equation, whereas the latter encompasses Lambert’s equation, among others. Both give rise to analytical formulae containing the coefficients of the original equation and are universally applicable; however, closed‐form solutions are expressible as a finite number of algebraic operations, whereas open‐form solutions require infinite series.Many equations do not hold an exact solution – either due to their intrinsic form, or because it was not yet found; in such cases, numerical methods are in order – which call for an initial search interval, or at least a single initial estimate. Only univariate problems will be discussed below in terms of numerical analysis; for more than one independent variable, such methods may be directly extended per se, and independently (although simultaneously) applied to each variable when searching for the overall root – whereas the partial derivative of the function under scrutiny, with regard to each such variable, is to be taken in the case of derivative‐dependent methods.7.1 Linear Systems of Equations
Consider a system of linear equations of the form(7.1)where xi 's (i = 1, 2, …, n) denote unknown variables, ai,j’s (i = 1, 2, …, m; j = 1, 2, …, n) denote (real) coefficients of said unknowns, and bi’s (i = 1, 2, …, m) denote independent terms; and further assume there is no constraint imposed upon the relative magnitude of (integer) m and n. Since the information to calculate the solution(s) is scattered through (and requires simultaneous consideration of) the whole set of equations as per Eq. (7.1) , one strategy is to calculate each variable on its own as a (linear) function of the remaining variables. For instance (and without loss of generality), one may solve the first equation for x1
