Show your work and check answer. -3x=5x+8
2 answers:
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Answer:
Step-by-step explanation:
As 2 and 7 are the only prime divisors of both and
we know that both can be written as:
Where is the set of all natural numbers adding the zero (careful because this part is important as I'll explain next).
We also know that 14 divides both numbers and that is actually the greatest common divisor between them. So we can rewrite a and b as follows:
Why do I write them like this? Because this way is easier to observe that if and
were both greater than zero, then 28 would divide both hence 14 wouldn't be their g.c.d.. Likewise, if
and
were both greater than zero, then 98 would divide both and once again, 14 wouldn't be their g.c.d.
So either of them has to be equal to zero. And then we have that
All we have left to do is find the possible values for and
so that
and that only happens if
and
Answer:
a) f(0)=4
b) f(-2)=2
c) x∈{-8,-4,4,8}
d) x∈{-9,-3,3,9}
e) x∈(-∞,-9)∪(-3,3)∪(9,∞).
f) x∈[-9,-3]∪[3,9]
g) Domain is x∈[-10,10]. Range is y∈[-4,4]. Zeros are at elements of x in {-9,-3,3,9}
h) Increasing on x∈ (-6,0) ∪ (6,10) and
decreasing on x∈ (-10,-6) ∪ (0,6).
Step-by-step explanation:
a) f(0) means what is the y-coordinate of the point at x=0.
So find 0 on the x-axis. I'm going to go up because the curve is at y=4 there.
Conclusion: f(0)=4.
b) f(-2) means what is the y-coordinate of the point at x=-2.
So find -2 on the x-axis. I'm going to go up because the curve is at y=2 there.
Conclusion: f(-2)=2.
c) f(x)=-2 means what is x when y=-2. So go on the y-axis and find -2. Now anything on that line y=-2 (horizontal line going through y=-2 on the y-axis) we need to look at.
There are 4 value we need to look at then.
x=-8
x=-4
x=4
x=8
At all of these the y-coordinate is -2.
d) f(x)=0 means what is x when y=0. So go on the y-axis and find 0. Now anything on that line y=0 (horizontal line going through y=0 on the y-axis; also knowing as the x-axis for y=0) we need to look at.
There are 4 values we need to look at then.
x=-9
x=-3
x=3
x=9
e) f(x)>0 means where is the curve above the x-axis.
The curve is above the x-axis:
- Before x=-9
- Between x=-3 and x=3
- After x=9
The interval notation is:
(-∞,-9)∪(-3,3)∪(9,∞).
d) f(x)≥0 means where is the curve below or on the x-axis (also known as y=0).
The curve is below or on the x-axis:
- Between -9 and -3 (inclusive of both endpoints because they include y=0).
- Between 3 and 9 (inclusive of both endpoints because they include y=0).
The interval notation is:
[-9,-3]∪[3,9]
g) The domain is where the curve exists for the x-values.
The domain is all real numbers between -10 and 10 (inclusive of both).
The curve starts at x=-10 and stops at x=10. The function exists a y value for any number between -10 and 10 (including both).
Interval notation is:
[-10,10]
The range is where the curve exists for the y-values.
Looking from bottom to top I see that it starts at y=-4 and stops at y=4. I notice the curve exists at some point between those two horizontal lines. It also exists at both of those endpoints" -4 and 4.
The range is between -4 and 4 (including both).
Interval notation is:
[-4,4]
The zeros are where the graph crosses the x-axis.
The graph crosses the x-axis at:
x=-9
x=-3
x=3
x=9
h) Reading left to right the graph increases when you see the curve going up.
I see this from x=-6 to x=0 (exclusive of both).
I see this from x=6 to x=10 (exclusive of both).
So interval notation is (-6,0) ∪ (6,10).
Reading left to right the graph decreases when you see the curve going down.
I see this from x=-10 to x=-6 ( exclusive of both).
I see this from x=0 to x=6 (exclusive of both).
