If r>0 and s>0, the solution to the inequality 3(x-r)^2>s will cover more of the number line when
A. r is increased
B. r is decreased
C. s is increased
D. s is decreased
the function is a parabola with its vertex at (r,0) and opening upward
If s <= 0, then the solution set is the whole number line.
If s > 0, then if s gets greater, less of the line is included.
So, the real answer is, of course, D
changing r just moves the vertex.
Well, well, well, let me put my funny hat on for this question! Looks like we have an inequality to solve, huh? 3(x-r)^2 > s, where r > 0 and s > 0. Now, the question is asking which variable, r or s, will make the solution cover more of the number line.
Let's think about this for a moment. If we increase r, that means we are subtracting a larger value from x, which will make the whole expression smaller. It's like taking a tiny little step backward on the number line. So, increasing r won't make the solution cover more of the number line.
Now, what happens if we decrease r? Well, in that case, we are subtracting an even smaller value from x, which means the whole expression becomes even bigger. It's like taking a big, long leap forward on the number line! So, by decreasing r, we can make the solution cover more of the number line.
What about s? If we increase s, well, that just means we're making the right side of the inequality bigger. It's like blowing a balloon, but without the fun squeaky noise. That won't really affect how much of the number line the solution covers.
Lastly, if we decrease s, we're making the right side of the inequality smaller. It's like deflating that balloon we just inflated - oh, the deflation drama! But again, decreasing s won't have much impact on how much of the number line the solution covers.
So, the answer is B. If we decrease r, we can make the solution cover more of the number line. Ah, the power of tiny steps! Keep calm, decrease r, and watch the solution dance across the number line!
To determine which option will cause the solution to the inequality 3(x-r)^2>s to cover more of the number line, let's analyze the equation.
Given: r > 0 and s > 0
We can start by expanding the expression (x-r)^2:
(x-r)^2 = (x-r)(x-r) = x^2 - 2rx + r^2
So the inequality becomes:
3(x^2 - 2rx + r^2) > s
Now, let's simplify the equation:
3x^2 - 6rx + 3r^2 > s
To find out how the solution will change with respect to r and s, we need to analyze the discriminant of the quadratic equation 3x^2 - 6rx + 3r^2 - s = 0.
The discriminant is calculated using the formula: b^2 - 4ac.
In this case, a = 3, b = -6r, and c = 3r^2 - s.
The discriminant is:
D = (-6r)^2 - 4(3)(3r^2 - s)
= 36r^2 - 4(3)(3r^2 - s)
= 36r^2 - 36(3r^2 - s)
= 36r^2 - 108r^2 + 36s
= -72r^2 + 36s
Now let's analyze the possible cases:
Case 1: If r is increased (option A)
Increasing the value of r will make -72r^2 smaller, shifting the discriminant D towards a positive value. This means that the inequality 3(x-r)^2 > s will cover less of the number line. Therefore, option A is incorrect.
Case 2: If r is decreased (option B)
Decreasing the value of r will make -72r^2 larger, shifting the discriminant D towards a negative value. This means that the inequality 3(x-r)^2 > s will cover more of the number line. Therefore, option B is correct.
Case 3: If s is increased (option C)
Increasing the value of s will make 36s larger, shifting the discriminant D towards a positive value. This means that the inequality 3(x-r)^2 > s will cover less of the number line. Therefore, option C is incorrect.
Case 4: If s is decreased (option D)
Decreasing the value of s will make 36s smaller, shifting the discriminant D towards a negative value. This means that the inequality 3(x-r)^2 > s will cover more of the number line. Therefore, option D is correct.
In conclusion, the solution to the inequality 3(x-r)^2 > s will cover more of the number line when option B (r is decreased) or option D (s is decreased) is chosen.
To determine which variable, r or s, affects the coverage of the number line when solving the inequality 3(x-r)^2 > s, we need to analyze how each variable affects the inequality.
Let's start by rephrasing the inequality using quadratic form:
3(x - r)(x - r) > s
Expanding the equation:
3(x^2 - 2rx + r^2) > s
Simplifying:
3x^2 - 6rx + 3r^2 > s
Now we can see that the quadratic equation is in the form of Ax^2 + Bx + C > 0, where A = 3, B = -6r, C = 3r^2 - s.
To solve this inequality, we need to consider the sign of the quadratic expression: 3x^2 - 6rx + 3r^2.
The graph of this quadratic equation is a parabola that opens upward (since the coefficient of x^2 is positive), and the sign of the quadratic expression depends on the values of A, B, and C.
For the quadratic expression to be greater than 0 (positive), we need it to be above the x-axis. This means we need either two positive x-intercepts or no x-intercepts.
To determine which variable affects the coverage of the number line, we need to consider the discriminant formula for quadriatic equations:
Discriminant (D) = B^2 - 4AC
If the discriminant is positive (D > 0), there are two x-intercepts, which means the solution will cover more of the number line.
If the discriminant is zero (D = 0), there is only one x-intercept, and the solution will cover less of the number line.
If the discriminant is negative (D < 0), there are no x-intercepts, and the solution will cover even less of the number line.
Now let's analyze the situation based on the variables r and s:
A. If r is increased, the coefficient B = -6r becomes more negative. Therefore, the discriminant D will increase, increasing the likelihood of having two x-intercepts. Thus, increasing r will cover more of the number line.
B. If r is decreased, the coefficient B = -6r becomes less negative. Therefore, the discriminant D will decrease, decreasing the likelihood of having two x-intercepts. Thus, decreasing r will cover less of the number line.
C. If s is increased, the coefficient C = 3r^2 - s becomes more negative. Since the discriminant D depends on the value of C, increasing s will make the quadratic expression more negative. As a result, the likelihood of having two x-intercepts will decrease. Thus, increasing s will cover less of the number line.
D. If s is decreased, the coefficient C = 3r^2 - s becomes less negative. As a result, the quadratic expression will be less negative, increasing the likelihood of having two x-intercepts. Thus, decreasing s will cover more of the number line.
Based on the above analysis, the correct answer is:
D. s is decreased