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Sagot :
Let's go through each problem step-by-step to determine the solutions.
1. Find the LCD of [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{2}{7}\)[/tex].
- The least common denominator (LCD) of 3 and 7 is [tex]\(21\)[/tex].
2. Find the LCD of [tex]\(\frac{3}{x-2}\)[/tex] and [tex]\(\frac{1}{x+3}\)[/tex].
- The LCD of [tex]\(x-2\)[/tex] and [tex]\(x+3\)[/tex] is [tex]\((x-2)(x+3)\)[/tex].
3. Find the sum of [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{2}{7}\)[/tex].
- We need to find a common denominator to add the fractions together. The LCD is 21.
- The sum is [tex]\(\frac{1}{3} + \frac{2}{7} = \frac{7}{21} + \frac{6}{21} = \frac{13}{21}\)[/tex].
4. Find the sum of [tex]\(\frac{2}{x} + \frac{5}{x}\)[/tex].
- The denominators are the same, so we can directly add the numerators: [tex]\(\frac{2+5}{x} = \frac{7}{x}\)[/tex].
5. Find the product of [tex]\(\frac{3}{8}\)[/tex] and [tex]\(\frac{12}{5}\)[/tex].
- Multiply the numerators and the denominators: [tex]\(\frac{3 \times 12}{8 \times 5} = \frac{36}{40} = \frac{9}{10}\)[/tex].
6. Find the sum of [tex]\(\frac{3}{x-2}\)[/tex] and [tex]\(\frac{1}{x+3}\)[/tex].
- To add these fractions, we need a common denominator, which is [tex]\((x-2)(x+3)\)[/tex].
- The sum is [tex]\(\frac{3(x+3) + 1(x-2)}{(x-2)(x+3)} = \frac{3x + 9 + x - 2}{(x-2)(x+3)} = \frac{4x+7}{x^2 + x - 6}\)[/tex].
7. Factor [tex]\(x^2 + x - 12\)[/tex].
- We look for two numbers that multiply to [tex]\(-12\)[/tex] and add up to [tex]\(1\)[/tex]. Those numbers are [tex]\(4\)[/tex] and [tex]\(-3\)[/tex].
- The factors are [tex]\((x + 4)(x - 3)\)[/tex].
Matching Column A with Column B, we get:
1. 21 (not listed in Column B explicitly)
2. E. [tex]\((x-2)(x+3)\)[/tex] or [tex]\(x^2 + x - 6\)[/tex]
3. I. [tex]\(\frac{13}{21}\)[/tex]
4. G. [tex]\(\frac{x+4}{x+2}\)[/tex] Note: Error - actually [tex]\(\frac{7}{x}\)[/tex]
5. Not correctly listed but would be [tex]\(\frac{9}{10}\)[/tex]
6. C. [tex]\(\frac{4x+7}{x^2 + x - 6}\)[/tex]
7. A. [tex]\((x+4)(x-3)\)[/tex]
Decoding the secret message by matching answers found:
1. (21 not listed, so likely no letter)
2. E
3. I
4. (Error in options, skip)
5. (Fraction equivalent missing, skip)
6. C
7. A
Secret message: _EIC_
1. Find the LCD of [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{2}{7}\)[/tex].
- The least common denominator (LCD) of 3 and 7 is [tex]\(21\)[/tex].
2. Find the LCD of [tex]\(\frac{3}{x-2}\)[/tex] and [tex]\(\frac{1}{x+3}\)[/tex].
- The LCD of [tex]\(x-2\)[/tex] and [tex]\(x+3\)[/tex] is [tex]\((x-2)(x+3)\)[/tex].
3. Find the sum of [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{2}{7}\)[/tex].
- We need to find a common denominator to add the fractions together. The LCD is 21.
- The sum is [tex]\(\frac{1}{3} + \frac{2}{7} = \frac{7}{21} + \frac{6}{21} = \frac{13}{21}\)[/tex].
4. Find the sum of [tex]\(\frac{2}{x} + \frac{5}{x}\)[/tex].
- The denominators are the same, so we can directly add the numerators: [tex]\(\frac{2+5}{x} = \frac{7}{x}\)[/tex].
5. Find the product of [tex]\(\frac{3}{8}\)[/tex] and [tex]\(\frac{12}{5}\)[/tex].
- Multiply the numerators and the denominators: [tex]\(\frac{3 \times 12}{8 \times 5} = \frac{36}{40} = \frac{9}{10}\)[/tex].
6. Find the sum of [tex]\(\frac{3}{x-2}\)[/tex] and [tex]\(\frac{1}{x+3}\)[/tex].
- To add these fractions, we need a common denominator, which is [tex]\((x-2)(x+3)\)[/tex].
- The sum is [tex]\(\frac{3(x+3) + 1(x-2)}{(x-2)(x+3)} = \frac{3x + 9 + x - 2}{(x-2)(x+3)} = \frac{4x+7}{x^2 + x - 6}\)[/tex].
7. Factor [tex]\(x^2 + x - 12\)[/tex].
- We look for two numbers that multiply to [tex]\(-12\)[/tex] and add up to [tex]\(1\)[/tex]. Those numbers are [tex]\(4\)[/tex] and [tex]\(-3\)[/tex].
- The factors are [tex]\((x + 4)(x - 3)\)[/tex].
Matching Column A with Column B, we get:
1. 21 (not listed in Column B explicitly)
2. E. [tex]\((x-2)(x+3)\)[/tex] or [tex]\(x^2 + x - 6\)[/tex]
3. I. [tex]\(\frac{13}{21}\)[/tex]
4. G. [tex]\(\frac{x+4}{x+2}\)[/tex] Note: Error - actually [tex]\(\frac{7}{x}\)[/tex]
5. Not correctly listed but would be [tex]\(\frac{9}{10}\)[/tex]
6. C. [tex]\(\frac{4x+7}{x^2 + x - 6}\)[/tex]
7. A. [tex]\((x+4)(x-3)\)[/tex]
Decoding the secret message by matching answers found:
1. (21 not listed, so likely no letter)
2. E
3. I
4. (Error in options, skip)
5. (Fraction equivalent missing, skip)
6. C
7. A
Secret message: _EIC_
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