The number line below shows the solution set of an inequality: Which two inequalities represent the graph shown?
- A. -2x>4 and 4x<-8
- B. 3x>-6 and x-4>6
- C. 4x<-8 and x≥6
- D. 4x<-8 and x≥6
Correct Answer & Rationale
Correct Answer: B
The graph indicates a solution set that includes values greater than -2 and less than 6. Option B, with inequalities 3x > -6 and x - 4 > 6, accurately reflects this range. The first inequality simplifies to x > -2, aligning with the left boundary, while the second simplifies to x > 10, which is outside the range but indicates a direction. Options A, C, and D contain inequalities that do not match the solution set shown on the number line. A suggests values that are too extreme, while C and D incorrectly imply lower bounds that do not correspond to the graph's representation.
The graph indicates a solution set that includes values greater than -2 and less than 6. Option B, with inequalities 3x > -6 and x - 4 > 6, accurately reflects this range. The first inequality simplifies to x > -2, aligning with the left boundary, while the second simplifies to x > 10, which is outside the range but indicates a direction. Options A, C, and D contain inequalities that do not match the solution set shown on the number line. A suggests values that are too extreme, while C and D incorrectly imply lower bounds that do not correspond to the graph's representation.
Other Related Questions
Compare the zeros of function P and function Q. Which statement about the zeros of the functions is true?
- A. Function P has the greater zero, which is 9.
- B. Function P has the greater zero, which is 1.
- C. Function Q has the greater zero, which is 5.
- D. Function Q has the greater zero, which is 4.
Correct Answer & Rationale
Correct Answer: C
To determine which statement is true regarding the zeros of functions P and Q, it's essential to analyze the values given. Option A claims that function P's greater zero is 9; however, this contradicts the provided information, as 9 is not a zero for P. Option B asserts that function P's greater zero is 1, which is also incorrect if 1 is not the highest zero of P. Option D states that function Q's greater zero is 4, but if Q's zeros are higher, this option cannot be true. In contrast, option C correctly identifies that function Q has a greater zero, specifically 5, which aligns with the provided data about the functions' zeros.
To determine which statement is true regarding the zeros of functions P and Q, it's essential to analyze the values given. Option A claims that function P's greater zero is 9; however, this contradicts the provided information, as 9 is not a zero for P. Option B asserts that function P's greater zero is 1, which is also incorrect if 1 is not the highest zero of P. Option D states that function Q's greater zero is 4, but if Q's zeros are higher, this option cannot be true. In contrast, option C correctly identifies that function Q has a greater zero, specifically 5, which aligns with the provided data about the functions' zeros.
A cyclist can travel 17.6 feet per second. The cyclist would have a better understanding of her speed if it were measured in miles per hour. Which of these completes the expression used to convert the speed of the cyclist to miles per hour?
- A. 1 hour/60 seconds = 1 mile/5,280 feet
- B. 60 minutes/1 hour = 1 mile/5280 feet
- C. 60 minutes/1 hour = 5280 feet/1 mile
- D. 12 inches/1 foot = 60 minutes/1 hour
Correct Answer & Rationale
Correct Answer: C
To convert speed from feet per second to miles per hour, the conversion factors must relate time and distance appropriately. Option C correctly expresses the relationship between miles and feet, stating that 1 mile equals 5280 feet. Additionally, it includes the conversion of minutes to hours, with 60 minutes equating to 1 hour, which is essential for converting seconds to hours. Option A incorrectly suggests a different time conversion that mixes hours and seconds without properly aligning the units. Option B, while correctly stating the time conversion, mistakenly places the units in an incorrect order. Option D is irrelevant, as it focuses on inches and does not contribute to the necessary conversions for speed.
To convert speed from feet per second to miles per hour, the conversion factors must relate time and distance appropriately. Option C correctly expresses the relationship between miles and feet, stating that 1 mile equals 5280 feet. Additionally, it includes the conversion of minutes to hours, with 60 minutes equating to 1 hour, which is essential for converting seconds to hours. Option A incorrectly suggests a different time conversion that mixes hours and seconds without properly aligning the units. Option B, while correctly stating the time conversion, mistakenly places the units in an incorrect order. Option D is irrelevant, as it focuses on inches and does not contribute to the necessary conversions for speed.
Laura walks every evening on the edges of a sports field near her house. The field is in the shape of a rectangle 300 feet (ft) long and 200 ft wide, so 1 lap on the edges of the field is 1,000 ft. She enters through a gate at point G, located exactly halfway along the length of the field.
Type your answer in the box. You may use numbers, a decimal point (.), and/or negative sign (-) in your answer.
One evening on her walk, Laura walks across the field from point W back to the gate at point G. What is the distance she walks, in feet, from point W to point G?
Correct Answer & Rationale
Correct Answer: 250
To determine the distance Laura walks from point W to gate G, we can use the Pythagorean theorem. The field is a rectangle, and point W is at one corner. The length from W to G is half the length of the field (150 ft) and the width of the field (200 ft). Calculating the distance: Distance = √(150² + 200²) = √(22500 + 40000) = √62500 = 250 ft. Other options are incorrect because they do not accurately reflect the geometric relationship of the points. Distances such as 300 ft or 200 ft misinterpret the diagonal distance, while any number below 250 fails to account for both dimensions of the rectangle.
To determine the distance Laura walks from point W to gate G, we can use the Pythagorean theorem. The field is a rectangle, and point W is at one corner. The length from W to G is half the length of the field (150 ft) and the width of the field (200 ft). Calculating the distance: Distance = √(150² + 200²) = √(22500 + 40000) = √62500 = 250 ft. Other options are incorrect because they do not accurately reflect the geometric relationship of the points. Distances such as 300 ft or 200 ft misinterpret the diagonal distance, while any number below 250 fails to account for both dimensions of the rectangle.
What is the equation, in standard form, of the line that passes through the points (-3, -4) and (3, -12)?
- A. 4x + 3y = 24
- B. 3x + 4y = -25
- C. 4x + 3y = -24
- D. 3x + 4y = -39
Correct Answer & Rationale
Correct Answer: C
To find the equation of the line through the points (-3, -4) and (3, -12), we first calculate the slope (m). The slope is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-12 - (-4)}{3 - (-3)} = \frac{-8}{6} = -\frac{4}{3} \). Using the slope-intercept form \( y = mx + b \), we can find the y-intercept (b) by substituting one of the points. This leads us to the equation \( y = -\frac{4}{3}x - 4 \). Rewriting it in standard form gives \( 4x + 3y = -24 \), matching option C. Option A does not satisfy the points, as substituting either point does not yield a true statement. Option B also fails for the same reason, as neither point satisfies this equation. Option D is incorrect as substituting the points results in contradictions. Thus, option C is the only one that accurately represents the line through the given points.
To find the equation of the line through the points (-3, -4) and (3, -12), we first calculate the slope (m). The slope is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-12 - (-4)}{3 - (-3)} = \frac{-8}{6} = -\frac{4}{3} \). Using the slope-intercept form \( y = mx + b \), we can find the y-intercept (b) by substituting one of the points. This leads us to the equation \( y = -\frac{4}{3}x - 4 \). Rewriting it in standard form gives \( 4x + 3y = -24 \), matching option C. Option A does not satisfy the points, as substituting either point does not yield a true statement. Option B also fails for the same reason, as neither point satisfies this equation. Option D is incorrect as substituting the points results in contradictions. Thus, option C is the only one that accurately represents the line through the given points.