Which equation represents a line that is positive in domain for each value of x.
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Which equation represents a line that is positive in domain for each value of x.
To find out if the equation intersects the x-axis, we need to substitute y=0 in each equation.
If the function has a solution where y=0 then the equation has an intersection point and is not the correct answer.
Let's start with the first equation:
y = 3x+8
We will substitute as instructed:
0 = 3x+8
3x = -8
x = -8/3
Although the result here is not a "nice" number, we see that we are able to arrive at a result and therefore this answer is rejected.
Let's move on to the second equation:
y = 300x+50
Here too we will substitute:
0 = 300x + 50
-50 = 300x
-50/300 = x
-1/6 = x
In this exercise too we managed to arrive at a result and therefore the answer is rejected.
Let's move on to answer C:
y = 3
We will substitute:
0 = 3
We see that here an impossible result is obtained because 0 can never be equal to 3.
Therefore, we understand that the equation in answer C is the one that does not intersect the x-axis, and is in fact positive all the time.
Therefore answer D is also rejected, and only answer C is correct.
Look at the linear function represented in the diagram.
When is the function positive?
Great question! While y = 3x + 8 has a positive slope (it's increasing), it still crosses the x-axis at . For x-values less than this, the function gives negative y-values.
Look for horizontal lines above the x-axis like y = 3, y = 5, etc. These never change value and never cross the x-axis. Any line with a slope (positive or negative) will eventually cross the x-axis somewhere.
An impossible equation like 0 = 3 means there's no solution! This is actually good news - it tells us the line never touches the x-axis, so it stays positive (or negative) for all x-values.
Yes! Horizontal lines below the x-axis like y = -2 or y = -10 are always negative. The same test works: setting y = 0 gives impossible equations like 0 = -2.
Setting y = 0 finds where the line crosses the x-axis. If we can solve for x, the line crosses and changes from positive to negative (or vice versa). If we can't solve (impossible equation), the line never crosses!
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