Studdy Solution
STEP 1
What is this asking?
For what values of x is 4x3+9x2 greater than or equal to 10x+3?
Watch out!
Don't forget to check your answer by plugging in values into the original inequality!
STEP 2
1. Rewrite the inequality
2. Find a root
3. Factor the cubic
4. Analyze the inequality
STEP 3
Let's **rewrite** this bad boy to make it easier to work with!
We want to get everything on one side, so let's subtract 10x and 3 from both sides of the inequality 4x3+9x2≥10x+3.
This gives us 4x3+9x2−10x−3≥0.
Now, we can **define** our function: f(x)=4x3+9x2−10x−3.
So, we're looking for when f(x)≥0.
STEP 4
From the graph, we see that x=−41 is a root.
This means f(−41)=0.
Let's **verify** this!
4(−41)3+9(−41)2−10(−41)−3=−644+169+410−3=−161+169+1640−1648=16−1+9+40−48=160=0
Woohoo! It works!
STEP 5
Since x=−41 is a root, we know (4x+1) is a factor of f(x).
Let's **perform polynomial long division** to find the other factor.
Dividing 4x3+9x2−10x−3 by 4x+1 gives us x2+2x−3.
So, we can **rewrite** f(x) as f(x)=(4x+1)(x2+2x−3).
STEP 6
Now, let's **factor** the quadratic term!
We're looking for two numbers that multiply to −3 and add to 2.
Those numbers are 3 and −1.
So, x2+2x−3=(x+3)(x−1).
This means f(x)=(4x+1)(x+3)(x−1).
STEP 7
We want to find when f(x)=(4x+1)(x+3)(x−1)≥0.
The roots are x=−41, x=−3, and x=1.
Let's **test** some values in the intervals determined by the roots!
STEP 8
* For x<−3 (e.g., x=−4), f(x) is negative.
* For −3<x<−41 (e.g., x=−1), f(x) is positive.
* For −41<x<1 (e.g., x=0), f(x) is negative.
* For x>1 (e.g., x=2), f(x) is positive.
STEP 9
So, f(x)≥0 when −3≤x≤−41 or x≥1.
STEP 10
The solution to the inequality 4x3+9x2≥10x+3 is x in the interval [−3,−41] or x in the interval [1,∞).
