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What is the rule for multiplying two rational numbers?
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When multiplying two rational numbers, multiply their numerators and denominators separately. If both numbers have the same sign, the product is positive; otherwise, it is negative.
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How do you divide one rational number by another?
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To divide by a rational number, multiply by its reciprocal. This means flipping the second fraction and then multiplying.
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What is the result when you multiply a negative and a positive rational number?
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The result is a negative rational number because a negative times a positive equals a negative.
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Explain the distributive property with a positive sign: $a(b+c)$.
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The distributive property states $a(b+c) = ab + ac$. You distribute $a$ to both $b$ and $c$.
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How does the distributive property work with a negative sign: $-a(b+c)$?
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The distributive property with a negative sign is $-a(b+c) = -ab - ac$. You distribute $-a$ to both $b$ and $c$, changing the sign of each term.
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Simplify the expression: $3x + 5x$.
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Combine like terms to get $8x$.
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What is the first step in combining like terms in the expression: $4y + 3 - 2y + 7$?
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Group the terms with the variable and the constants separately: $(4y - 2y) + (3 + 7)$.
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Identify and factor out the greatest common factor of $15xy$ and $25x$.
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The GCF is $5x$, so factor it out to get $5x(3y + 5)$.
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How do you identify like terms in an algebraic expression?
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Like terms have the same variable raised to the same power. For example, $2x$ and $3x$ are like terms.
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Provide an example of a real-world problem involving multiplying rational numbers.
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If a recipe requires $2/3$ cup of sugar and you want to make it triple, multiply to find the amount: $3 \times (2/3) = 2$ cups of sugar.
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What happens when you divide a positive rational number by a negative rational number?
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The result is negative because a positive divided by a negative equals a negative.
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Simplify: $-2(3x - 4)$.
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Use the distributive property: $-2 \times 3x + (-2) \times (-4) = -6x + 8$.
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What is a rational number?
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A rational number is any number that can be expressed as the quotient or fraction $p/q$ of two integers, where $q \neq 0$.
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Find the result of $\frac{3}{4} \div \frac{5}{6}$.
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Multiply by the reciprocal: $\frac{3}{4} \times \frac{6}{5} = \frac{18}{20} = \frac{9}{10}$ after simplifying.
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When distributing a negative number across terms in parentheses, what should you be cautious about?
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Ensure that the sign of each term in the parentheses changes after distribution.
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Give an example of factoring out the greatest common factor in a real-world problem.
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If two ribbons are $24$ inches and $36$ inches long, and need equal pieces, factor the GCF, $12$, so each piece is $12$ inches.
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What is the result of $(-1) \times (-8)$?
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The result is positive $8$, because the product of two negative numbers is positive.
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How can you use rational numbers to solve a problem involving percentages?
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Convert the percentage to a fraction and multiply by the total. E.g., $25\%$ of $80$ is $0.25 \times 80 = 20$.
Why is it important to understand operations with rational numbers for real-world applications?
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Understanding operations with rational numbers is important for tasks such as cooking, budgeting, and measurement conversions where precise calculations are required.
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