1.1 rational and irrational numbers
write each fraction as a decimal or each decimal as a fraction.
1.

1.27

3.
solve each equation for x.
4.
5.
6.

a square patio has an area of 200 square feet. how long is each side of the patio to the nearest tenth?

1.2 sets of real numbers
write all names that apply to each number.
8.
9.

tell whether the statement “all integers are rational numbers” is true or false. explain your choice.

1.3 ordering real numbers
compare. write <, >, or =.
11.
12.
order the numbers from least to greatest.
13.
14.

how are real numbers used to describe real - world situations?

Question

1.1 rational and irrational numbers
write each fraction as a decimal or each decimal as a fraction.
1.
2. 1.27
3.
solve each equation for x.
4.
5.
6.
7. a square patio has an area of 200 square feet. how long is each side of the patio to the nearest tenth?
1.2 sets of real numbers
write all names that apply to each number.
8.
9.
10. tell whether the statement “all integers are rational numbers” is true or false. explain your choice.

1.3 ordering real numbers
compare. write <, >, or =.
11.
12.
order the numbers from least to greatest.
13.
14.
15. how are real numbers used to describe real - world situations?

Accepted Answer

### Problem 1: $\boldsymbol{\frac{1}{3}}$ as a decimal # Explanation: ## Step1: Divide 1 by 3 $1 \div 3 = 0.\overline{3}$ (repeating decimal) # Answer: $0.\overline{3}$ ### Problem 2: $\boldsymbol{1.\overline{27}}$ as a fraction # Explanation: ## Step1: Let $x = 1.\overline{27}$ $x = 1.272727\ldots$ ## Step2: Multiply by 100 (2 repeating digits) $100x = 127.272727\ldots$ ## Step3: Subtract $x$ from $100x$ $100x - x = 127.2727\ldots - 1.2727\ldots$ $99x = 126$ ## Step4: Solve for $x$ $x = \frac{126}{99} = \frac{14}{11}$ (simplify by dividing numerator/denominator by 9) # Answer: $\frac{14}{11}$ ### Problem 3: $\boldsymbol{1\frac{5}{8}}$ as a decimal # Explanation: ## Step1: Convert mixed number to improper fraction $1\frac{5}{8} = \frac{13}{8}$ ## Step2: Divide 13 by 8 $13 \div 8 = 1.625$ # Answer: $1.625$ ### Problem 4: Solve $\boldsymbol{x^4 = 81}$ for $x$ # Explanation: ## Step1: Take 4th root of both sides $x = \pm \sqrt[4]{81}$ ## Step2: Simplify $\sqrt[4]{81}$ $\sqrt[4]{81} = 3$ (since $3^4 = 81$) # Answer: $x = \pm 3$ ### Problem 5: Solve $\boldsymbol{x^3 = 343}$ for $x$ # Explanation: ## Step1: Take cube root of both sides $x = \sqrt[3]{343}$ ## Step2: Simplify $\sqrt[3]{343}$ $\sqrt[3]{343} = 7$ (since $7^3 = 343$) # Answer: $x = 7$ ### Problem 6: Solve $\boldsymbol{x^2 = \frac{1}{100}}$ for $x$ # Explanation: ## Step1: Take square root of both sides $x = \pm \sqrt{\frac{1}{100}}$ ## Step2: Simplify square root $\sqrt{\frac{1}{100}} = \frac{1}{10}$ # Answer: $x = \pm \frac{1}{10}$ ### Problem 7: Side length of square patio (area = 200 sq ft) # Explanation: ## Step1: Area of square = $s^2$, so $s = \sqrt{ ext{Area}}$ $s = \sqrt{200}$ ## Step2: Simplify $\sqrt{200}$ $\sqrt{200} = \sqrt{100 imes 2} = 10\sqrt{2} \approx 14.142$ ## Step3: Round to nearest tenth $14.1$ (since $0.142 < 0.15$, round down) # Answer: $14.1$ feet ### Problem 8: Classify $\boldsymbol{\frac{121}{\sqrt{121}}}$ # Explanation: ## Step1: Simplify denominator $\sqrt{121} = 11$ ## Step2: Simplify fraction $\frac{121}{11} = 11$ ## Step3: Classify 11 11 is an integer, rational number, real number. # Answer: Integer, Rational, Real ### Problem 9: Classify $\boldsymbol{\frac{\pi}{2}}$ # Explanation: $\pi$ is irrational, so $\frac{\pi}{2}$ is also irrational (irrational ÷ rational = irrational). It is also a real number. # Answer: Irrational, Real ### Problem 10: “All integers are rational numbers” (True/False) # Brief Explanations: A rational number is $\frac{a}{b}$ where $a, b$ are integers and $b eq 0$. Any integer $n$ can be written as $\frac{n}{1}$ (e.g., $5 = \frac{5}{1}$), so it satisfies the definition of a rational number. # Answer: True. All integers can be expressed as $\frac{ ext{integer}}{1}$, so they are rational. ### Problem 11: Compare $\boldsymbol{\sqrt{8} + 3}$ and $\boldsymbol{8 + \sqrt{3}}$ # Explanation: ## Step1: Approximate square roots $\sqrt{8} \approx 2.828$, $\sqrt{3} \approx 1.732$ ## Step2: Calculate both expressions $\sqrt{8} + 3 \approx 2.828 + 3 = 5.828$ $8 + \sqrt{3} \approx 8 + 1.732 = 9.732$ ## Step3: Compare $5.828 < 9.732$ # Answer: $\sqrt{8} + 3 \boldsymbol{<} 8 + \sqrt{3}$ ### Problem 12: Compare $\boldsymbol{\sqrt{5} + 11}$ and $\boldsymbol{5 + \sqrt{11}}$ # Explanation: ## Step1: Approximate square roots $\sqrt{5} \approx 2.236$, $\sqrt{11} \approx 3.317$ ## Step2: Calculate both expressions $\sqrt{5} + 11 \approx 2.236 + 11 = 13.236$ $5 + \sqrt{11} \approx 5 + 3.317 = 8.317$ ## Step3: Compare $13.236 > 8.317$ # Answer: $\sqrt{5} + 11 \boldsymbol{>} 5 + \sqrt{11}$ ### Problem 13: Order $\boldsymbol{\sqrt{99}}$, $\boldsymbol{\pi}$, $\boldsymbol{9.8}$ (least to greatest) # Explanation: ## Step1: Approximate values $\sqrt{99} \approx 9.949$, $\pi \approx 3.141$, $9.8 = 9.8$ ## Step2: Order from least to greatest $\pi \approx 3.141 < 9.8 < \sqrt{99} \approx 9.949$ # Answer: $\pi$, $9.8$, $\sqrt{99}$ ### Problem 14: Order $\boldsymbol{\sqrt{\frac{1}{25}}}$, $\boldsymbol{\frac{1}{4}}$, $\boldsymbol{0.2}$ (least to greatest) # Explanation: ## Step1: Simplify each number $\sqrt{\frac{1}{25}} = \frac{1}{5} = 0.2$ $\frac{1}{4} = 0.25$ $0.2 = 0.2$ ## Step2: Order from least to greatest $0.2 = \sqrt{\frac{1}{25}} < \frac{1}{4}$ (since $0.2 < 0.25$) # Answer: $0.2$, $\sqrt{\frac{1}{25}}$, $\frac{1}{4}$ (or $0.2 = \sqrt{\frac{1}{25}} < \frac{1}{4}$) ### Problem 15: Real numbers in real - world situations # Brief Explanations: Real numbers (rational/irrational) describe quantities like: - **Length/Area**: Patio side length (Problem 7), room dimensions. - **Money**: Prices, budgets (e.g., $\$1.50 = \frac{3}{2}$). - **Time**: 2.5 hours (rational), or $\pi$ hours (theoretical). - **Measurements**: Temperature ($-5^\circ$C, $22.7^\circ$C), weight (3.14 kg). They allow precise description of continuous quantities (e.g., distance, time) and discrete ones (e.g., counts as integers). # Answer: Real numbers describe continuous (e.g., length, time) and discrete (e.g., counts) quantities. Examples: measuring a patio’s side (irrational $\sqrt{200}$), pricing ($1.27$), or temperature ($-5$). They enable precise real - world measurements.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1: $\boldsymbol{\frac{1}{3}}$ as a decimal

Step1: Divide 1 by 3

Step1: Let $x = 1.\overline{27}$

Step2: Multiply by 100 (2 repeating digits)

Step3: Subtract $x$ from $100x$

Step4: Solve for $x$

Step1: Convert mixed number to improper fraction

Step2: Divide 13 by 8

Answer:

Problem 2: $\boldsymbol{1.\overline{27}}$ as a fraction