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
The width of a painting is 24 centimeters shorter than its length, x. The area of the painting is 4,081 square centimeters. Which equation could be used to find the dimensions of the painting?
- A. x^2 - 24x - 4,081 = 0
- B. x^2 + 24x - 4,081 = 0
- C. x^2 + 24x + 4,081 = 0
- D. x^2 - 24x + 4,081 = 0
Correct Answer & Rationale
Correct Answer: A
To find the dimensions of the painting, we start with the relationship between length and width. The width is 24 cm shorter than the length \(x\), so it can be expressed as \(x - 24\). The area of a rectangle is given by the product of its length and width, resulting in the equation \(x(x - 24) = 4,081\). Expanding this leads to \(x^2 - 24x - 4,081 = 0\), which matches option A. Option B incorrectly adds 24x, leading to an incorrect area calculation. Option C incorrectly adds 24 and includes a positive constant, which does not represent the area. Option D incorrectly adds 4,081 and has a positive term that does not reflect the relationship between length and width.
To find the dimensions of the painting, we start with the relationship between length and width. The width is 24 cm shorter than the length \(x\), so it can be expressed as \(x - 24\). The area of a rectangle is given by the product of its length and width, resulting in the equation \(x(x - 24) = 4,081\). Expanding this leads to \(x^2 - 24x - 4,081 = 0\), which matches option A. Option B incorrectly adds 24x, leading to an incorrect area calculation. Option C incorrectly adds 24 and includes a positive constant, which does not represent the area. Option D incorrectly adds 4,081 and has a positive term that does not reflect the relationship between length and width.
Simplify 6^2 - 3^2
- A. 6
- B. 9
- C. 27
- D. 3
Correct Answer & Rationale
Correct Answer: C
To simplify \(6^2 - 3^2\), we apply the difference of squares formula, which states \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = 6\) and \(b = 3\). Thus, we have: \[ 6^2 - 3^2 = (6 - 3)(6 + 3) = 3 \times 9 = 27 \] Option A (6) is incorrect as it miscalculates the expression. Option B (9) mistakenly considers only one of the squared terms. Option D (3) misinterprets the operations involved, leading to an incorrect result. The correct evaluation yields 27, confirming option C as the accurate answer.
To simplify \(6^2 - 3^2\), we apply the difference of squares formula, which states \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = 6\) and \(b = 3\). Thus, we have: \[ 6^2 - 3^2 = (6 - 3)(6 + 3) = 3 \times 9 = 27 \] Option A (6) is incorrect as it miscalculates the expression. Option B (9) mistakenly considers only one of the squared terms. Option D (3) misinterprets the operations involved, leading to an incorrect result. The correct evaluation yields 27, confirming option C as the accurate answer.
Factor the expression completely: 45bcx - 10ax
- A. 5x(9bc - 2a)
- B. 5(9bc - 2a)
- C. x(45bc - 10a)
- D. 5x(9bc + 2a)
Correct Answer & Rationale
Correct Answer: A
To factor the expression 45bcx - 10ax completely, we start by identifying the greatest common factor (GCF). The GCF of the coefficients 45 and 10 is 5, and both terms contain the variable x. Thus, we can factor out 5x, resulting in 5x(9bc - 2a). Option A accurately reflects this factorization. Option B lacks the variable x, which is essential in the original expression. Option C incorrectly factors out only x, missing the GCF of 5. Option D alters the sign of the second term, which does not represent the original expression correctly.
To factor the expression 45bcx - 10ax completely, we start by identifying the greatest common factor (GCF). The GCF of the coefficients 45 and 10 is 5, and both terms contain the variable x. Thus, we can factor out 5x, resulting in 5x(9bc - 2a). Option A accurately reflects this factorization. Option B lacks the variable x, which is essential in the original expression. Option C incorrectly factors out only x, missing the GCF of 5. Option D alters the sign of the second term, which does not represent the original expression correctly.
A figure is formed by shaded squares on a grid. Which figure has a perimeter of 12units and an area of 8 square units?
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A.
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B.
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C.
- D. None of the above
Correct Answer & Rationale
Correct Answer: C
To determine the figure that meets the criteria of having a perimeter of 12 units and an area of 8 square units, we analyze each option. Option C achieves both requirements: it has a perimeter of 12 units, calculated by adding the lengths of all sides, and an area of 8 square units, determined by multiplying its length and width (2 x 4). In contrast, Option A has a perimeter exceeding 12 units, while its area is less than 8 square units. Option B has a perimeter of 10 units and an area of 6 square units, failing both criteria. Option D is not applicable since Option C meets the conditions. Thus, Option C stands out as the only figure that satisfies both the perimeter and area requirements.
To determine the figure that meets the criteria of having a perimeter of 12 units and an area of 8 square units, we analyze each option. Option C achieves both requirements: it has a perimeter of 12 units, calculated by adding the lengths of all sides, and an area of 8 square units, determined by multiplying its length and width (2 x 4). In contrast, Option A has a perimeter exceeding 12 units, while its area is less than 8 square units. Option B has a perimeter of 10 units and an area of 6 square units, failing both criteria. Option D is not applicable since Option C meets the conditions. Thus, Option C stands out as the only figure that satisfies both the perimeter and area requirements.