Inequalities Maths
In mathematics, inequalities refer to mathematical expressions that compare two quantities and indicate their relative sizes. These expressions use symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Inequalities are used to represent relationships between numbers and are fundamental in solving equations and making comparisons.
5 Key excerpts on "Inequalities Maths"
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...10 SOLVING 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 problem SECTIONS 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 Inequality A 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...
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...For example, since 4 < 8, it is also true that • Multiplication and division property of inequalities An 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 that An 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 −2 x < 6, then SOLVING 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 – 2 x ≤ 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 is a set of real numbers where any number that lies between two numbers in the set is also in the same set. An interval can be represented in more than one way, as shown in the accompanying table. If an interval does not include an endpoint value, it is “open” on that side...
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...So if a < 7, 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. 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. 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. Example 5.23. 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? Answer 5.23. The inequality to use is sales × commission ≥ $1,000. In this case, let’s let sales = x, and the inequality becomes 40 x ≥ 1000...
eBook - ePub- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Remember to check the answer in the original equation since squaring both sides can introduce extraneous solutions, solutions that are incorrect. So we need to check our solutions in the original equation. EXAMPLE Solve for x :. SOLUTION 5 x − 6 = 9 5 x = 15 x = 3 Check:. EXAMPLE Solve for x :. SOLUTION 2 x + − = 16 2 x = 14 x = 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 to When 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...
eBook - ePubGMAT Advanced Quant
250+ Practice Problems & Online Resources
- (Author)
- 2020(Publication Date)
- Manhattan Prep(Publisher)
...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 < −3 x + 5<22 x 2 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...
