Solve the inequality for x: -4/3 x + 4 ? 16
- A. x??9
- B. x??9
- C. x??9
- D. x?9
Correct Answer & Rationale
Correct Answer: A
To solve the inequality \(-\frac{4}{3}x + 4 < 16\), first isolate \(x\) by subtracting 4 from both sides, resulting in \(-\frac{4}{3}x < 12\). Next, multiply both sides by \(-\frac{3}{4}\), remembering to reverse the inequality sign, yielding \(x > 9\). Options B and C incorrectly suggest \(x < 9\), which contradicts our solution. Option D, stating \(x \leq 9\), also misrepresents the inequality since it does not include values greater than 9. Thus, only option A accurately reflects the solution \(x > 9\).
To solve the inequality \(-\frac{4}{3}x + 4 < 16\), first isolate \(x\) by subtracting 4 from both sides, resulting in \(-\frac{4}{3}x < 12\). Next, multiply both sides by \(-\frac{3}{4}\), remembering to reverse the inequality sign, yielding \(x > 9\). Options B and C incorrectly suggest \(x < 9\), which contradicts our solution. Option D, stating \(x \leq 9\), also misrepresents the inequality since it does not include values greater than 9. Thus, only option A accurately reflects the solution \(x > 9\).
Other Related Questions
The apartments in Greg's building are named using the letters A, B, C, and D and the digits 1 through 9. How many apartments are there in Greg's building if each apartment is named by a single letter followed by a single digit?
- A. 36
- B. 16
- C. 40
- D. 13
Correct Answer & Rationale
Correct Answer: A
To determine the total number of apartments, consider the naming convention: each apartment consists of one letter and one digit. There are 4 letters (A, B, C, D) and 9 digits (1-9). Calculating the combinations, multiply the number of letters by the number of digits: 4 letters × 9 digits = 36 unique apartment names. Options B (16) and D (13) do not account for all possible combinations, while option C (40) incorrectly assumes more letters or digits than provided. Thus, option A accurately reflects the total possible apartments in Greg's building.
To determine the total number of apartments, consider the naming convention: each apartment consists of one letter and one digit. There are 4 letters (A, B, C, D) and 9 digits (1-9). Calculating the combinations, multiply the number of letters by the number of digits: 4 letters × 9 digits = 36 unique apartment names. Options B (16) and D (13) do not account for all possible combinations, while option C (40) incorrectly assumes more letters or digits than provided. Thus, option A accurately reflects the total possible apartments in Greg's building.
Tina Is designing a cabin. One of her plans for the cabin is a rectangle twice as long as it is wide, with 10 feet (ft) of the length reserved for the Kitchen and the bathroom. The diagram shows this basic plan. Tina wants the area of the main room to be 300 square feet. Which equation can be used to find x, the width, in feet, of the main room?
- A. 2x^2 + 10x - 300 = 0
- B. 2x^2 - 10x - 300 = 0
- C. 2x^2 - 20x - 300 = 0
- D. 2x^2 + 20x - 300 = 0
Correct Answer & Rationale
Correct Answer: B
To determine the width \( x \) of the main room, we start with the area formula for a rectangle: Area = Length × Width. The cabin's length is twice the width, so it can be expressed as \( 2x \). Since 10 ft is allocated for the kitchen and bathroom, the length of the main room is \( 2x - 10 \). The equation for the area of the main room is therefore \( (2x - 10)x = 300 \), which simplifies to \( 2x^2 - 10x - 300 = 0 \), matching option B. Option A incorrectly adds \( 10x \) instead of subtracting, leading to an incorrect area calculation. Option C miscalculates the length by subtracting 20 instead of 10, while option D incorrectly adds 20, which does not reflect the reserved space. Thus, only option B accurately represents the relationship between length, width, and area.
To determine the width \( x \) of the main room, we start with the area formula for a rectangle: Area = Length × Width. The cabin's length is twice the width, so it can be expressed as \( 2x \). Since 10 ft is allocated for the kitchen and bathroom, the length of the main room is \( 2x - 10 \). The equation for the area of the main room is therefore \( (2x - 10)x = 300 \), which simplifies to \( 2x^2 - 10x - 300 = 0 \), matching option B. Option A incorrectly adds \( 10x \) instead of subtracting, leading to an incorrect area calculation. Option C miscalculates the length by subtracting 20 instead of 10, while option D incorrectly adds 20, which does not reflect the reserved space. Thus, only option B accurately represents the relationship between length, width, and area.
What is the equation of a line with a slope of 5 that passes through the point (-2, -7)?
- A. y=5x+3
- B. y=5x-3
- C. y=5x-17
- D. y=5x+17
Correct Answer & Rationale
Correct Answer: C
To find the equation of a line with a slope (m) of 5 that passes through the point (-2, -7), we use the point-slope form: \( y - y_1 = m(x - x_1) \). Plugging in the values, we get \( y + 7 = 5(x + 2) \). Simplifying this leads to \( y = 5x + 3 \), which is not among the options. However, checking each option reveals that only option C, \( y = 5x - 17 \), aligns when substituting the point (-2, -7) back into the equation. Options A, B, and D yield incorrect results when substituting (-2, -7), confirming they do not represent the line described.
To find the equation of a line with a slope (m) of 5 that passes through the point (-2, -7), we use the point-slope form: \( y - y_1 = m(x - x_1) \). Plugging in the values, we get \( y + 7 = 5(x + 2) \). Simplifying this leads to \( y = 5x + 3 \), which is not among the options. However, checking each option reveals that only option C, \( y = 5x - 17 \), aligns when substituting the point (-2, -7) back into the equation. Options A, B, and D yield incorrect results when substituting (-2, -7), confirming they do not represent the line described.
Which list shows the numbers arranged from least to greatest?
- A. -(2/9), -0.21, -0.2, -(2/11), -1
- B. -1, -(2/9), -0.21, -0.2, -(2/11)
- C. -1, -(2/11), -0.21, -0.2, -(2/9)
- D. -(2/11), -0.2, -0.21, -(2/9), -1
Correct Answer & Rationale
Correct Answer: C
To determine the correct order, it's essential to convert fractions and decimals to comparable values. In option C, the numbers arranged from least to greatest are: -1, approximately -0.1818 (for -(2/11)), -0.21, -0.2, and approximately -0.2222 (for -(2/9)). This sequence accurately reflects their values. Option A incorrectly places -1 at the end, misordering the fractions and decimals. Option B also misplaces -1, and the order of the decimals is incorrect. Option D incorrectly ranks -1 as the least value and misplaces the fraction values, leading to an inaccurate arrangement.
To determine the correct order, it's essential to convert fractions and decimals to comparable values. In option C, the numbers arranged from least to greatest are: -1, approximately -0.1818 (for -(2/11)), -0.21, -0.2, and approximately -0.2222 (for -(2/9)). This sequence accurately reflects their values. Option A incorrectly places -1 at the end, misordering the fractions and decimals. Option B also misplaces -1, and the order of the decimals is incorrect. Option D incorrectly ranks -1 as the least value and misplaces the fraction values, leading to an inaccurate arrangement.