Quadratic Inequalities: solve 10-9y≥2y^2 and a^2 ≤4(2a-3)

[shaznayel]

How do you solve 10-9y≥2y^2 and a^2 ≤4(2a-3) ?

 

[Otis]

How do you solve 10-9y≥2y^2?

Start by solving the equality part. Then, we can discuss the inequality part.

10 - 9y = 2y^2

If you find that you need help, while solving this quadratic equation, please show us how far you got.

Or, in case you don't know how to solve a quadratic equation, let us know that, and we can provide links to online lessons/videos.

 

[shaznayel]

10-9y≥2y^2
10-9y=2y^2
-2y^2-9y+10=0
1[-2y^2-9y+10=0]
2y^2+9y-10=0
so using the quadratic equation, I got -9 plus the square root of 161 over 4 and -9 minus the square root of 161 over 4.
Is that right?

 

[Otis]

10-9y ≥ 2y^2

10-9y ≥ 2y^2

-2y^2-9y+10 ≥ 0

-1[-2y^2-9y+10 ≥ 0]

2y^2+9y-10 ≤ 0

so using the quadratic equation, I got -9 plus the square root of 161 [all] over 4

and

-9 minus the square root of 161 [all] over 4.

Is that right?

Except for a minor typo shown in red above, that's correct!

We can type your solutions for the equality part like this:

y = [-9 + sqrt(161)]/4

or

y = [-9 - sqrt(161)]/4

Now, I was going to discuss the inequality part in terms of a graph, but I changed my mind because that would require renaming the horizontal axis as the y-axis. I don't want to confuse, so I ought to have done it algebraically from the start.

Therefore, I have changed all your equal signs back to inequality symbols. Note that the inequality symbol changed direction when you multiplied both sides by -1. (You knew to do that, yes?)

2y^2+9y-10 ≤ 0

Now, you found the two values of y where 2y^2+9y-10 equals zero.

Using a scientific calculator, we can roughly approximate these values as -5.4 and 0.9. These values divide the Real number line into three intervals:

All numbers less than -5.4

All numbers inbetween -5.4 and 0.9

All numbers greater than 0.9

You need to determine the interval(s) where the value of 2y^2+9y-10 is less than zero.

So, pick a test value for y from each interval, and use it to evaluate the expression 2y^2+9y-10. If your test value results in a negative value for 2y^2+9y-10, then ALL values in that interval cause 2y^2+9y-10 to be negative.

The interval(s) where 2y^2+9y-10 is less than zero give the solutions for the original inequality.