Suitable range of values for x

  • Thread starter Richie Smash
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In summary, the conversation discusses a function and its corresponding x and y values, as well as using a graph to determine the range of values of x for a given inequality. The method of graphing and solving a simple quadratic equation is suggested to find the range of values.
  • #1
Richie Smash
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Homework Statement


There is a function y= 3+x-2x2

X values are as follows: -3, -2, -1, 0, 1, 2, 3.
Y values are as follows:-18, -7, 0, 3, 2, -3,-12.

Each x value corresponds to the y value below it after going through the function.

I have plotted a graph using a scale of 1 cm on the Y axis and a scale of 2 cm on the X axis, I have obtained a Parabola opening downwards.

Now the Question says, '' Using your graph or otherwise, determine the range of values of x for which x -2x2>-6.

Homework Equations

The Attempt at a Solution



Now, I guess I can do this by substituing every x value into that inequality... but is there a faster way to do this

But in doing so I got -2 is less than or equal to x which is less than or equal to 2
Also how can I look at my graph and know this?
 
Last edited:
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  • #2
Try plotting y = x - 2x2 and compare to the original
 
  • #3
Ok, I have done that and obtained another downwards facing parabola, underneath the previous one.
 
  • #4
Here is a picture.
 

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  • #5
Richie Smash said:

Homework Statement


There is a function y= 3+x-2x2

X values are as follows: -3, -2, -1, 0, 1, 2, 3.
Y values are as follows:-18, -7, 0, 3, 2,-3,-12.

Each x value corresponds to the y value below it after going through the function.

I have plotted a graph using a scale of 1 cm on the Y axis and a scale of 2 cm on the X axis, I have obtained a Parabola opening downwards.

Now the Question says, '' Using your graph or otherwise, determine the range of values of x for which x -2x2>-6.

Homework Equations

The Attempt at a Solution



Now, I guess I can do this by substituing every x value into that inequality... but is there a faster way to do this

But in doing so I got -2 is less than or equal to x which is less than or equal to 2
Also how can I look at my graph and know this?

The graph is a parabola opening downward, so for very large |x| (i.e, large positive or large negative) the graph
Richie Smash said:

Homework Statement


There is a function y= 3+x-2x2

X values are as follows: -3, -2, -1, 0, 1, 2, 3.
Y values are as follows:-18, -7, 0, 3, 2,-3,-12.

Each x value corresponds to the y value below it after going through the function.

I have plotted a graph using a scale of 1 cm on the Y axis and a scale of 2 cm on the X axis, I have obtained a Parabola opening downwards.

Now the Question says, '' Using your graph or otherwise, determine the range of values of x for which x -2x2>-6.

Homework Equations

The Attempt at a Solution



Now, I guess I can do this by substituing every x value into that inequality... but is there a faster way to do this

But in doing so I got -2 is less than or equal to x which is less than or equal to 2
Also how can I look at my graph and know this?

Since the parabola opens downward, then for large positive x or large negative x the graph of ##y =x - 2 x^2## will lie below the line ##y = -6##. Therefore, there must be some lowest value ##x_1## and largest value ##x_2## of ##x## that gives ##x - 2 x^2 >-6##. These will occur where ##x - 2 x^2 = -6##. This is a simple quadratic equation, which you can solve using the usual "quadratic root" formula. There are two roots: the smaller root will be ##x_1## and the larger root will be ##x_2##.
 
  • #6
Hi Richie Smash!
So this is your graph. The dots are the given values you wrote above.
"Determine the range of values of ##x## for which ##x-2x^{2}>-6##"
You can see that ##x-2x^{2}>-6## means ##y>-3## (just plus 3 both side, nothing change)
Look at your graph, find every ##x## values which make ##y>-3## ##\leftarrow## just draw the line ##y=-3 ## then see which values ##x## "above" ##y=-3## line (I see 3 points)
Attention, do not obtain values ##x## where ##y=-3## because ##x-2x^{2}>-6##, not ##x-2x^{2}\geq-6##
That's the way you can use your graph, hope my guide can help you :biggrin:
 

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  • #7
Ray Vickson said:
The graph is a parabola opening downward, so for very large |x| (i.e, large positive or large negative) the graphSince the parabola opens downward, then for large positive x or large negative x the graph of ##y =x - 2 x^2## will lie below the line ##y = -6##. Therefore, there must be some lowest value ##x_1## and largest value ##x_2## of ##x## that gives ##x - 2 x^2 >-6##. These will occur where ##x - 2 x^2 = -6##. This is a simple quadratic equation, which you can solve using the usual "quadratic root" formula. There are two roots: the smaller root will be ##x_1## and the larger root will be ##x_2##.

Nguyen Son said:
Hi Richie Smash!
So this is your graph. The dots are the given values you wrote above.
"Determine the range of values of ##x## for which ##x-2x^{2}>-6##"
You can see that ##x-2x^{2}>-6## means ##y>-3## (just plus 3 both side, nothing change)
Look at your graph, find every ##x## values which make ##y>-3## ##\leftarrow## just draw the line ##y=-3 ## then see which values ##x## "above" ##y=-3## line (I see 3 points)
Attention, do not obtain values ##x## where ##y=-3## because ##x-2x^{2}>-6##, not ##x-2x^{2}\geq-6##
That's the way you can use your graph, hope my guide can help you :biggrin:

Hi Ray vickson, I solved the quadratic and obtained
x= -1.5
x= 2

Hi Nyugen Son, I'm not quite sure about your method?
 
Last edited:
  • #8
I understand the solving the Quadratic equation, but How do I check my answers on the graph to determine the range?
 
  • #9
Try graphing y = -6 on top of your parabola graph
 
  • #10
When I then drew the line y = -6, I found the range which matches up with the quadratic equation I solved for the parabola x-2x2.

However the range with the line y= -6 for the parabola y= 3+x-2x2 was not the correct range.

I was only when I drew the line y= -3 did the range for the parabola y= 3+x-2x that the range was identical to the one found using the line y=-6 for the parabola x-2x2.

I think I understand now as I can see the ranges clearly on my graph, what I don't understand is why does changing y = -6 to y = -3 make the original parabola also satisfy the inequality?
 
  • #11
Richie Smash said:
I understand the solving the Quadratic equation, but How do I check my answers on the graph to determine the range?

You don't need to (but doing so cannot hurt).

The smallest root ##x_1 = -1.5## is a dividing point between the regions where ##x -2x^2 < -6## and ##x - 2 x^2 > -6##. Since for large negative ##x## we have ##x - 2x^2 < -6## (parabola below the line at -6), all points to the left of ##x_1## give ##x - 2x^2 < -6## and points a bit to the right of ##x_1## have ##x - 2x^2 > - 6##. That continues to happen until ##x## grows large enough to hit ##x_2 = 2##, which is the second dividing point between ##x - 2x^2 > -6## and ##x - 2 x^2 < -6##. Therefore, all points between ##x_1## and ##x_2## satisfy the required inequality.
 
Last edited:

1. What is the suitable range of values for x?

The suitable range of values for x depends on the specific equation or problem being solved. It can range from real numbers (-∞, ∞) to integers (-∞, ∞), or it can be limited to a specific interval or set of numbers. It is important to carefully read and understand the problem in order to determine the suitable range of values for x.

2. How do I determine the suitable range of values for x?

To determine the suitable range of values for x, you should consider any given constraints or restrictions in the problem. This could be in the form of inequalities, limits, or specific values that x cannot take. You can also plot the equation or problem on a graph to visually determine the range of values for x.

3. Can the suitable range of values for x be negative?

Yes, the suitable range of values for x can include negative numbers. It all depends on the problem and the specific values of x that satisfy it. For example, in a quadratic equation, x can have both positive and negative values.

4. Is there a maximum or minimum value for x in the suitable range?

It depends on the specific problem and its constraints. In some cases, x may have a maximum or minimum value, while in others, it may be an open-ended range of values. Again, carefully reading and understanding the problem is crucial in determining the maximum or minimum value for x.

5. Can the suitable range of values for x be an infinite set?

Yes, the suitable range of values for x can be an infinite set, such as all real numbers or all positive integers. This is common in many mathematical and scientific equations and problems. It is important to consider the context of the problem to determine if an infinite set is a valid range of values for x.

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