Answer :
To solve the problem, we need to choose a number from the provided list that will make both given expressions irrational. We need to evaluate each number to determine if it satisfies this condition. Let's consider our given list of numbers: [tex]\(3.2, \sqrt{3}, \frac{2}{3}, \text{and } 0.232323\ldots \)[/tex].
First, let's transform the two expressions by substituting the chosen number [tex]\( x \)[/tex]:
1. Expression 1: [tex]\(-\frac{5}{9} + x\)[/tex]
2. Expression 2: [tex]\(4 + x\)[/tex]
To determine which number from the list makes both expressions irrational, let's evaluate each one:
1. For [tex]\( x = 3.2 \)[/tex]:
- Expression 1: [tex]\(-\frac{5}{9} + 3.2 = -\frac{5}{9} + \frac{32}{10} = -\frac{5}{9} + \frac{288}{90} = \frac{288-50}{90} = \frac{238}{90} = \frac{119}{45}\)[/tex] (which is rational)
- Expression 2: [tex]\(4 + 3.2 = 7.2\)[/tex] (which is rational)
Since both results are rational, 3.2 is not a solution.
2. For [tex]\( x = \sqrt{3} \)[/tex]:
- Expression 1: [tex]\(-\frac{5}{9} + \sqrt{3} \)[/tex]
Since [tex]\(\sqrt{3}\)[/tex] is irrational, and any irrational number remains irrational when added to a rational number ( [tex]\(-\frac{5}{9}\)[/tex] is rational), [tex]\(-\frac{5}{9} + \sqrt{3}\)[/tex] is irrational.
- Expression 2: [tex]\(4 + \sqrt{3}\)[/tex]
Similar reasoning: Adding an irrational number ([tex]\(\sqrt{3}\)[/tex]) to a rational number (4) results in an irrational number.
Since both results are irrational, [tex]\(\sqrt{3}\)[/tex] is a solution.
3. For [tex]\( x = \frac{2}{3} \)[/tex]:
- Expression 1: [tex]\(-\frac{5}{9} + \frac{2}{3} = -\frac{5}{9} + \frac{6}{9} = \frac{1}{9}\)[/tex] (which is rational)
- Expression 2: [tex]\(4 + \frac{2}{3} = 4.666\)[/tex] (which is rational)
Since both results are rational, [tex]\(\frac{2}{3}\)[/tex] is not a solution.
4. For [tex]\( x = 0.232323\ldots \)[/tex] (which is a repeating decimal, or [tex]\(\frac{23}{99}\)[/tex]):
- Expression 1: [tex]\(-\frac{5}{9} + \frac{23}{99}\)[/tex]
This summation might seem complex, but important is whether any sum of rational numbers can produce an irrational number. Adding two rational numbers results in a rational number. Hence, [tex]\(-\frac{5}{9} + \frac{23}{99}\)[/tex] is rational.
- Expression 2: [tex]\(4 + 0.232323\ldots = 4.232323\ldots\)[/tex] (which is rational)
Since both results are rational, [tex]\(0.232323\ldots\)[/tex] is not a solution.
Hence, the only number from the list that makes both expressions irrational is [tex]\(\sqrt{3}\)[/tex].
Answer: [tex]\(\sqrt{3}\)[/tex]
Explanation:
When [tex]\(\sqrt{3}\)[/tex] is substituted into both expressions:
- [tex]\(-\frac{5}{9} + \sqrt{3}\)[/tex] results in an irrational number because adding an irrational number ([tex]\(\sqrt{3}\)[/tex]) to a rational number results in an irrational number.
- [tex]\(4 + \sqrt{3}\)[/tex] also results in an irrational number for the same reason.
Thus, [tex]\(\sqrt{3}\)[/tex] is the correct choice.
First, let's transform the two expressions by substituting the chosen number [tex]\( x \)[/tex]:
1. Expression 1: [tex]\(-\frac{5}{9} + x\)[/tex]
2. Expression 2: [tex]\(4 + x\)[/tex]
To determine which number from the list makes both expressions irrational, let's evaluate each one:
1. For [tex]\( x = 3.2 \)[/tex]:
- Expression 1: [tex]\(-\frac{5}{9} + 3.2 = -\frac{5}{9} + \frac{32}{10} = -\frac{5}{9} + \frac{288}{90} = \frac{288-50}{90} = \frac{238}{90} = \frac{119}{45}\)[/tex] (which is rational)
- Expression 2: [tex]\(4 + 3.2 = 7.2\)[/tex] (which is rational)
Since both results are rational, 3.2 is not a solution.
2. For [tex]\( x = \sqrt{3} \)[/tex]:
- Expression 1: [tex]\(-\frac{5}{9} + \sqrt{3} \)[/tex]
Since [tex]\(\sqrt{3}\)[/tex] is irrational, and any irrational number remains irrational when added to a rational number ( [tex]\(-\frac{5}{9}\)[/tex] is rational), [tex]\(-\frac{5}{9} + \sqrt{3}\)[/tex] is irrational.
- Expression 2: [tex]\(4 + \sqrt{3}\)[/tex]
Similar reasoning: Adding an irrational number ([tex]\(\sqrt{3}\)[/tex]) to a rational number (4) results in an irrational number.
Since both results are irrational, [tex]\(\sqrt{3}\)[/tex] is a solution.
3. For [tex]\( x = \frac{2}{3} \)[/tex]:
- Expression 1: [tex]\(-\frac{5}{9} + \frac{2}{3} = -\frac{5}{9} + \frac{6}{9} = \frac{1}{9}\)[/tex] (which is rational)
- Expression 2: [tex]\(4 + \frac{2}{3} = 4.666\)[/tex] (which is rational)
Since both results are rational, [tex]\(\frac{2}{3}\)[/tex] is not a solution.
4. For [tex]\( x = 0.232323\ldots \)[/tex] (which is a repeating decimal, or [tex]\(\frac{23}{99}\)[/tex]):
- Expression 1: [tex]\(-\frac{5}{9} + \frac{23}{99}\)[/tex]
This summation might seem complex, but important is whether any sum of rational numbers can produce an irrational number. Adding two rational numbers results in a rational number. Hence, [tex]\(-\frac{5}{9} + \frac{23}{99}\)[/tex] is rational.
- Expression 2: [tex]\(4 + 0.232323\ldots = 4.232323\ldots\)[/tex] (which is rational)
Since both results are rational, [tex]\(0.232323\ldots\)[/tex] is not a solution.
Hence, the only number from the list that makes both expressions irrational is [tex]\(\sqrt{3}\)[/tex].
Answer: [tex]\(\sqrt{3}\)[/tex]
Explanation:
When [tex]\(\sqrt{3}\)[/tex] is substituted into both expressions:
- [tex]\(-\frac{5}{9} + \sqrt{3}\)[/tex] results in an irrational number because adding an irrational number ([tex]\(\sqrt{3}\)[/tex]) to a rational number results in an irrational number.
- [tex]\(4 + \sqrt{3}\)[/tex] also results in an irrational number for the same reason.
Thus, [tex]\(\sqrt{3}\)[/tex] is the correct choice.
