Math  /  Algebra

QuestionSolve the inequality. (4x9)4(x1)3(x+1)0(4 x-9)^{4}(x-1)^{3}(x+1) \leq 0 (,1][1,94](-\infty,-1] \cup\left[1, \frac{9}{4}\right] (,1]{94}(-\infty, 1] \cup\left\{\frac{9}{4}\right\} [1,1]{94}[-1,1] \cup\left\{\frac{9}{4}\right\} (,1][1,)(-\infty,-1] \cup[1, \infty) [1,1][94,)[-1,1] \cup\left[\frac{9}{4}, \infty\right)

Studdy Solution

STEP 1

What is this asking? We need to find the values of xx that make this complicated inequality with exponents true! Watch out! Don't forget to consider the impact of those exponents on the sign of each factor, and remember zero is our friend here!

STEP 2

1. Analyze the factors
2. Determine signs
3. Combine and conquer

STEP 3

Let's **break down** this inequality into smaller, manageable pieces.
We have three main factors: (4x9)4(4x - 9)^4, (x1)3(x - 1)^3, and (x+1)(x + 1).

STEP 4

Notice that (4x9)4(4x - 9)^4 is raised to an **even power**.
This means it will *always* be greater than or equal to zero, regardless of the value of xx.
The only time it *equals* zero is when 4x9=04x - 9 = 0, which means x=94x = \frac{9}{4}.
Keep this in mind!

STEP 5

Now, let's look at (x1)3(x - 1)^3.
This factor has an **odd power**, so it can be positive, negative, *or* zero.
It's zero when x=1x = 1, positive when x>1x > 1, and negative when x<1x < 1.

STEP 6

Finally, we have (x+1)(x + 1).
This one's simple!
It's zero when x=1x = -1, positive when x>1x > -1, and negative when x<1x < -1.

STEP 7

Let's **visualize** the signs of each factor on a number line.
We'll use our **critical values** of xx: 1-1, 11, and 94\frac{9}{4}.

STEP 8

When x<1x < -1, we have (4x9)4(4x - 9)^4 is positive, (x1)3(x - 1)^3 is negative, and (x+1)(x + 1) is negative.
Multiplying these signs together (positive times negative times negative) gives us a **positive** result.

STEP 9

When 1<x<1-1 < x < 1, we have (4x9)4(4x - 9)^4 is positive, (x1)3(x - 1)^3 is negative, and (x+1)(x + 1) is positive.
Multiplying these signs gives us a **negative** result.

STEP 10

When 1<x<941 < x < \frac{9}{4}, we have (4x9)4(4x - 9)^4 is positive, (x1)3(x - 1)^3 is positive, and (x+1)(x + 1) is positive.
Multiplying these signs gives us a **positive** result.

STEP 11

When x>94x > \frac{9}{4}, all three factors are positive, resulting in a **positive** product.

STEP 12

We're looking for values of xx where the entire expression is *less than or equal to* zero.
From our sign analysis, we know the expression is negative when 1<x<1-1 < x < 1.

STEP 13

Don't forget the "or equal to" part!
The expression is equal to zero when any of the factors are zero.
This happens at x=1x = -1, x=1x = 1, and x=94x = \frac{9}{4}.

STEP 14

Putting it all together, our solution is the interval [1,1][-1, 1] (including the endpoints) and the single value x=94x = \frac{9}{4}.

STEP 15

The solution to the inequality (4x9)4(x1)3(x+1)0(4x - 9)^4(x - 1)^3(x + 1) \leq 0 is [1,1]{94}[-1, 1] \cup \{\frac{9}{4}\}.

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