Summarize how graphs can be used to find solutions to polynomial inequalities.

Question

huckelig75 · Accepted Answer

Step 1
1. If necessary, rewrite the inequality so that there is a zero on one side.
2. Graph the other side of the inequality.
3. Use your graphing calculator's TRACE function to the exact or approximate values of the x-intercepts.
4. Find the intervals that satisfy the inequality.
5. Write the final result as a solution interval.
Step 2
Example:
$x^{3} - 4 x^{2} + x + 5 \leq 3$
1. We substract 3 from both sides of the inequality:
$x^{3} - 4 x^{2} + x + 2 \leq 0$
2. We graph $y = x^{3} - 4 x^{2} + x + 2$

Step 3
With the help of the graphing calculator we find the x-intercepts as

x \in {- 0.562, 1, 3.562}

, rounded to three decimal digits:

Step 4
4. We note that the inequality is satisfied on the intervals

(- \infty, - 0.562]

and

1, 3.562]

.
5. We write the result as a solution interval:

{x \in R ∣ x \leq - 0.562 or 1 \leq x \leq 3.562}

or alternatively

x \in (- \infty, - 0.562] \cup [1, 3.562]

Provere · Accepted Answer

Step 1
Q2 has been attached here for reference:
$V o l u m e = 2 x^{3} - 30 x^{2} + 100 x$
$2 x^{3} - 30 x^{2} + 100 x \geq 72$

Step 2
The x-axis of the graph of a function is the number line over which the function is changing and therefore can be read to find the domain of x for different ranges of the value of

f (x)

.

Summarize how graphs can be used to find solutions to polynomial inequalities.

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