-2x^2<30
1/abs value(11-9x)=6
Please show me how to do these two!!
Sorry, Ellie, I'm not quite sure what it is you have to do with them - graph them? specify a range for which they are defined, or true in the first case? make a table of values?
Anyway, I'll talk a bit about them.
The first one says -2 * x^2 < 30. Now, this one is obviously true for all x. x^2 must be a positive number, or zero right? So -2x^2 must be negative or zero for all x. Certainly, it's always going to be less than 30. We can say that this is true for all real values of x. The graph of -2x^2 will be a parabola rising to touch the origin and falling down again.
We can write 1/|11-9x|=6 as 1/|11-9x|-6=0.
First consider 11-9x. As x gets very big or very negative, 11-9x gets even more so, since 9x dominates. |11-9x| will also get big, but only in the positive direction. So, far away from x=0, the number keeps increasing. But as |11-9x| keeps increasing, 1/|11-9x| will obviously keep shrinking closer and closer to zero, but staying positive, so 1/|11-9x|-6 will be almost a straight line with a value just above 6, getting closer to 6 as x gets bigger or smaller.
At the point x=11/9, 11-9x becomes 11 - 9* 11/9 = zero, so |11-9x| is zero, so the result is undefined. But |11-9x| always stays positive, so the value of the expression rises asymptotically as x approaches 11/9. So the graph will be almost a straight line along y=-6 except near the point x=11/9, where it will curve upward steeply, off to infinity, from both sides of the point.
I hope this helps.
-2x^2<30
x^2 > -15
Now think about our last statement.
Isn't the square of any number positive, and isn't any positive number greater than -15 (or any negative) ?
So the solution is the set of all real numbers.
For the second, let's cross-multiply to get
│11-9x│ = 1/6
then +(11-9x) = 1/6 OR -(11-9x) = 1/6
for +(11-9x) = 1/6
66 - 54x = 1
54x = 65
x = 65/54 = 13/9
for -(11-9x) = 1/6
-66 + 54x = 1
54x = 67
x = 67/54
My bad, Ellie and Reiny, on the second one. My thinker slipped a cog, and I considered y = 1/|11-9x|-6 rather than 1/|11-9x|-6=0. Reiny is correct, of course!
To solve the inequality -2x^2 < 30, we follow these steps:
1. Divide both sides by -2 to isolate x^2:
-2x^2 / -2 > 30 / -2
x^2 > -15
2. We notice that the inequality involves a quadratic expression. To solve quadratic inequalities, we can first set x^2 - 15 = 0 and solve for the values of x that make this equation true.
3. By rearranging the equation, we have:
x^2 = 15
4. Take the square root of both sides:
x = ±√15
So, the solution to the quadratic inequality -2x^2 < 30 is x < -√15 or x > √15.
Moving on to the second equation 1/|11-9x| = 6:
1. Multiply both sides of the equation by the absolute value of the expression:
|11 - 9x| = 1/6
2. Since the absolute value of a quantity can be positive or negative, we need to consider both cases.
Case 1:
Set the expression inside the absolute value equal to the positive value (1/6):
11 - 9x = 1/6
Case 2:
Set the expression inside the absolute value equal to the negative value (-1/6):
11 - 9x = -1/6
3. Solve each case for x:
Case 1:
11 - 9x = 1/6
11 - 1/6 = 9x
66/6 - 1/6 = 9x
65/6 = 9x
x = 65/54
Case 2:
11 - 9x = -1/6
11 + 1/6 = 9x
66/6 + 1/6 = 9x
67/6 = 9x
x = 67/54
Therefore, the solutions to the equation 1/|11-9x| = 6 are x = 65/54 and x = 67/54.