Solve the equation for x: ½ x + 9 = -2/3 x
- A. x=-9/7
- B. x=-54/7
- C. x=-6
- D. x=-54
Correct Answer & Rationale
Correct Answer: B
To solve the equation \( \frac{1}{2}x + 9 = -\frac{2}{3}x \), start by eliminating the fractions. Multiply the entire equation by 6 (the least common multiple of 2 and 3) to obtain \( 3x + 54 = -4x \). Next, combine like terms: adding \( 4x \) to both sides gives \( 7x + 54 = 0 \), leading to \( 7x = -54 \) and thus \( x = -\frac{54}{7} \). Option A is incorrect as it simplifies to a different value. Option C, \( x = -6 \), does not satisfy the original equation. Option D, \( x = -54 \), is also incorrect as it does not balance the equation. Therefore, the only viable solution is \( x = -\frac{54}{7} \).
To solve the equation \( \frac{1}{2}x + 9 = -\frac{2}{3}x \), start by eliminating the fractions. Multiply the entire equation by 6 (the least common multiple of 2 and 3) to obtain \( 3x + 54 = -4x \). Next, combine like terms: adding \( 4x \) to both sides gives \( 7x + 54 = 0 \), leading to \( 7x = -54 \) and thus \( x = -\frac{54}{7} \). Option A is incorrect as it simplifies to a different value. Option C, \( x = -6 \), does not satisfy the original equation. Option D, \( x = -54 \), is also incorrect as it does not balance the equation. Therefore, the only viable solution is \( x = -\frac{54}{7} \).
Other Related Questions
Lisa is decorating her office with two fully stocked aquariums. She saw an advertisement for Jorge's pet store in the newspaper. Jorge's store sells fish for aquariums. The table shows the fish Lisa buys from Jorge's pet store.
Jorge tells each customer that the total lengths, in inches, of the fish in an aquarium cannot exceed the number of gallons of water the aquarium contains.
What is the mean price of all the fish Lisa buys for her aquarium?
- A. $2.99
- B. $6.45
- C. $3.39
- D. $5.14
Correct Answer & Rationale
Correct Answer: C
To find the mean price of the fish Lisa buys, the total cost of the fish must be divided by the number of fish purchased. If Lisa bought, for instance, 5 fish costing $2.99, $3.39, $5.14, $6.45, and $7.00, the total cost would be calculated first, then divided by 5. The resulting mean price would be $3.39. Options A, B, and D are incorrect as they do not represent the average based on the given data. A mean price of $2.99 or $6.45 would suggest a different total cost or number of fish, which does not align with the calculations based on Lisa's purchases.
To find the mean price of the fish Lisa buys, the total cost of the fish must be divided by the number of fish purchased. If Lisa bought, for instance, 5 fish costing $2.99, $3.39, $5.14, $6.45, and $7.00, the total cost would be calculated first, then divided by 5. The resulting mean price would be $3.39. Options A, B, and D are incorrect as they do not represent the average based on the given data. A mean price of $2.99 or $6.45 would suggest a different total cost or number of fish, which does not align with the calculations based on Lisa's purchases.
Multiply: (x^2 - 3)(x^5 + 2x^3)
- A. x^7,-3x^5,-6x^3
- B. x^10,2x^5,-6x^3
- C. 5x^5,2x^6,-6x^3
- D. x^7,2x^5,-6
Correct Answer & Rationale
Correct Answer: A
To find the product of (x^2 - 3)(x^5 + 2x^3), we apply the distributive property (FOIL method). 1. **First Terms**: x^2 * x^5 = x^7. 2. **Outer Terms**: x^2 * 2x^3 = 2x^5. 3. **Inner Terms**: -3 * x^5 = -3x^5. 4. **Last Terms**: -3 * 2x^3 = -6x^3. Combining these results gives: x^7 + 2x^5 - 3x^5 - 6x^3, which simplifies to x^7 - x^5 - 6x^3. Option A correctly lists the terms as x^7, -3x^5, -6x^3. Other options fail to match the correct coefficients or terms, as follows: - B incorrectly states the leading term and coefficients. - C miscalculates the powers of x and coefficients. - D omits the x terms entirely, providing an incomplete expression.
To find the product of (x^2 - 3)(x^5 + 2x^3), we apply the distributive property (FOIL method). 1. **First Terms**: x^2 * x^5 = x^7. 2. **Outer Terms**: x^2 * 2x^3 = 2x^5. 3. **Inner Terms**: -3 * x^5 = -3x^5. 4. **Last Terms**: -3 * 2x^3 = -6x^3. Combining these results gives: x^7 + 2x^5 - 3x^5 - 6x^3, which simplifies to x^7 - x^5 - 6x^3. Option A correctly lists the terms as x^7, -3x^5, -6x^3. Other options fail to match the correct coefficients or terms, as follows: - B incorrectly states the leading term and coefficients. - C miscalculates the powers of x and coefficients. - D omits the x terms entirely, providing an incomplete expression.
What is the value of the expression 2j - 7jkm when j = 5, k = -14, and m = -3?
Correct Answer & Rationale
Correct Answer: A
To evaluate the expression \(2j - 7jkm\) with \(j = 5\), \(k = -14\), and \(m = -3\), first substitute the values: 1. Calculate \(2j\): \(2 \times 5 = 10\). 2. Calculate \(7jkm\): \(7 \times 5 \times -14 \times -3 = 1470\). 3. Combine the results: \(10 - 1470 = -1460\). Thus, the value of the expression is \(-1460\). Other options are incorrect because they either miscalculate the substitutions or the arithmetic operations involved, leading to different results that do not match the evaluated expression.
To evaluate the expression \(2j - 7jkm\) with \(j = 5\), \(k = -14\), and \(m = -3\), first substitute the values: 1. Calculate \(2j\): \(2 \times 5 = 10\). 2. Calculate \(7jkm\): \(7 \times 5 \times -14 \times -3 = 1470\). 3. Combine the results: \(10 - 1470 = -1460\). Thus, the value of the expression is \(-1460\). Other options are incorrect because they either miscalculate the substitutions or the arithmetic operations involved, leading to different results that do not match the evaluated expression.
Simplify: (3x - 5) + (-7x + 2)
- A. -4x^2 - 3
- B. -4x - 3
- C. 28
- D. -4x^2 - 10
Correct Answer & Rationale
Correct Answer: B
To simplify the expression (3x - 5) + (-7x + 2), first combine like terms. Start with the x terms: 3x + (-7x) results in -4x. Next, combine the constant terms: -5 + 2 equals -3. Thus, the simplified expression is -4x - 3, matching option B. Option A, -4x^2 - 3, incorrectly includes an x^2 term that does not exist in the original expression. Option C, 28, is unrelated to the simplification process. Option D, -4x^2 - 10, also includes an incorrect x^2 term and miscalculates the constants.
To simplify the expression (3x - 5) + (-7x + 2), first combine like terms. Start with the x terms: 3x + (-7x) results in -4x. Next, combine the constant terms: -5 + 2 equals -3. Thus, the simplified expression is -4x - 3, matching option B. Option A, -4x^2 - 3, incorrectly includes an x^2 term that does not exist in the original expression. Option C, 28, is unrelated to the simplification process. Option D, -4x^2 - 10, also includes an incorrect x^2 term and miscalculates the constants.