Compound inequalities that make use of the logical and require that all inequalities are solved by a single solution. The solution set is the intersection of each individual solution set. Compound inequalities of the form dark n A m can be decomposed into two inequalities using the logical and. However, it is just as valid to consider the argument A to be bounded between the values n and. Topic Exercises Part A: Simple Inequalities Graph all solutions on a number line and provide the corresponding interval notation. x 34 Part B: Compound Inequalities Graph all solutions on a number line and give the corresponding interval notation. 2 x. 0 x 50.
To describe compound inequalities such as x 3 or x6, write xx 3 or x6, which is read the set of all real numbers x such that x is less than 3 or x is greater than or equal. Write bounded intervals, such as 1x 3, as x1x 3, which is read the set of all real numbers x such that x is greater than or equal to dream 1 and less than. Key takeaways Inequalities usually have infinitely many solutions, so rather than presenting an impossibly large list, we present such solutions sets either graphically on a number line or textually using interval notation. Inclusive inequalities with the or equal to component are indicated with a closed dot on the number line and with a square bracket using interval notation. Strict inequalities without the or equal to component are indicated with an open dot on the number line and a parenthesis using interval notation. Compound inequalities that make use of the logical or are solved by solutions of either inequality. The solution set is the union of each individual solution set.
We have used set notation to list the elements such as the integers The braces group the elements of the set and the ellipsis marks indicate that the integers continue forever. In this section, we wish to describe intervals of real numbers—for example, the real numbers greater than or equal. Since the set is too large to list, set-builder notation allows us to describe it using familiar mathematical notation. An example of set-builder notation follows: Here x r describes the type of number, where the symbol is read element. This implies that the variable x represents a real number. The vertical bar is read such that. Finally, the statement x2 is the condition that describes the set using mathematical notation. At this point in our study of algebra, it is assumed that all variables represent real numbers. For this reason, you can omit the r and write xx2, which is read the set of all real numbers x such that x is greater than or equal.
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For example, one such solution. Notice that 1 is between banking 1 and 3 on a number line, or that 1. Similarly, we can see that other possible solutions are 1,.99, 0,.0056,.8, and.99. Since there are infinitely many real numbers between 1 and 3, we must express the solution graphically and/or with interval notation, in this case 1, 3). Example 6: Graph and give the interval notation equivalent: 32. Solution: Shade all real numbers bounded by, or strictly between, 32112 and.
Answer: Interval notation: (32, 2) Example 7: Graph and give the interval notation equivalent: 5 x15. Solution: Shade all real numbers between 5 and 15, and indicate story that the upper bound, 15, is included in the solution set by using a closed dot. Answer: Interval notation: (5, 15 In the previous two examples, we did not decompose the inequalities; instead we chose to think of all real numbers between the two given bounds. In summary, set-builder Notation In this text, we use interval notation. However, other resources that you are likely to encounter use an alternate method for describing sets called set-builder notation A system for describing sets using familiar mathematical notation.
Solution: Combine all solutions of both inequalities. The solutions to each inequality are sketched above the number line as a means to determine the union, which is graphed on the number line below. Answer: Interval notation: 3 any real number less than 3 in the shaded region on the number line will satisfy at least one of the two given inequalities. Example 4: Graph and give the interval notation equivalent: x 3 or x1. Solution: Both solution sets are graphed above the union, which is graphed below.
Answer: Interval notation: r ) When you combine both solution sets and form the union, you can see that all real numbers satisfy the original compound inequality. In summary, and bounded Intervals An inequality such as reads 1 one is less than or equal to x and x is less than three. This is a compound inequality because it can be decomposed as follows: The logical and requires that both conditions must be true. Both inequalities are satisfied by all the elements in the intersection The set formed by the shared values of the individual solution sets that is indicated by the logical use of the word and, denoted with the symbol., denoted, of the solution sets of each. Example 5: Graph and give the interval notation equivalent: x 3 and x1. Solution: Determine the intersection, or overlap, of the two solution sets. The solutions to each inequality are sketched above the number line as a means to determine the intersection, which is graphed on the number line below. Here x3 is not a solution because it solves only one of the inequalities. Answer: Interval notation: 1, 3) Alternatively, we may interpret 1x 3 as all possible values for x between or bounded by 1 and 3 on a number line.
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For example, 5 can be tree expressed textually as (inf,. Two inequalities save in one statement joined by the word and or by the word. Is actually two or more inequalities in one statement joined by the word and or by the word. Compound inequalities with the logical or require that either condition must be satisfied. Therefore, the solution set of this type of compound inequality consists of all the elements of the solution sets of each inequality. When we join these individual solution sets it is called the union, the set formed by joining the individual solution sets indicated by the logical use of the word or and denoted with the symbol., denoted. For example, the solutions to the compound inequality x 3 or x6 can be graphed as follows: Sometimes we encounter compound inequalities where the separate solution sets overlap. In the case where the compound inequality contains the word or, we combine all the elements of both sets to create one set containing all the elements of each. Example 3: Graph and give the interval notation equivalent: x1 or.
Use negative infinity, the energy symbol indicates the interval is unbounded to the left. to indicate that the solution set is unbounded to the left on a number line. Answer: Interval notation: 3 example 2: Graph and give the interval notation equivalent:. Solution: Use a closed dot and shade all numbers less than and including. Answer: Interval notation: 5, it is important to see that 5x is the same. Both require values of x to be smaller than or equal. To avoid confusion, it is good practice to rewrite all inequalities with the variable on the left. Also, when using text, use inf as a shortened form of infinity.
a real number: it cannot be included in the solution set. Now compare the interval notation in the previous example to that of the strict, or noninclusive, inequality that follows: Strict inequalities, express ordering relationships using the symbol for less than and for greater than. Imply that solutions may get very close to the boundary point, in this case 2, but not actually include. Denote this idea with an open dot on the number line and a round parenthesis in interval notation. Example 1: Graph and give the interval notation equivalent:. Solution: Use an open dot at 3 and shade all real numbers strictly less than.
To express the solution graphically, draw a number line and shade in all the values that essay are solutions to the inequality. Interval notation is textual and uses specific notation as follows: Determine the interval notation after graphing the solution set on a number line. The numbers in interval notation should be written in the same order as they appear on the number line, with smaller numbers in the set appearing first. In this example, there is an inclusive inequality, an inequality that includes the boundary point indicated by the or equal part of the symbols and and a closed dot on the number line., which means that the lower-bound 2 is included in the solution. Denote this with a closed dot on the number line and a square bracket in interval notation. The symbol is read as infinity, the symbol indicates the interval is unbounded to the right. And indicates that the set is unbounded to the right on a number line.
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Unbounded Intervals, an algebraic inequality, expressions related with the symbols, and., such as x2, is read x is greater than or equal. This inequality has infinitely many solutions for. Some of the solutions are 2, 3,.5, 5, 20, and.001. Since it is impossible to list all writing of the solutions, a system is needed that allows a clear communication of this infinite set. Two common ways of expressing solutions to an inequality are by graphing them on a number line. Solutions to an algebraic inequality expressed by shading the solution on a number line. And using interval notation, a textual system of expressing solutions to an algebraic inequality.