Math / Algebra

QuestionNeed Help? Read it Submit Answer 11. [0/1 Points] DETAILS MY NOTES SPRECALC7 1.11.037.
Find the solutions of the inequality by drawing appropriate graphs. State the answer rounded to two decimals. (Enter your answer using interval notation.) $\begin{array}{c} x^{1 / 3}<x \\ (-1,0) \cup(0,1) \end{array}$

Studdy Solution

STEP 1

1. We need to solve the inequality $x^{1/3} < x$ .
2. We will use graphical methods to find the solutions.
3. The solutions should be stated in interval notation and rounded to two decimal places.

STEP 2

1. Graph the function $y = x^{1/3}$ .
2. Graph the function $y = x$ .
3. Find the points of intersection of the graphs.
4. Determine the intervals where $x^{1/3} < x$ based on the graphs.
5. Write the solution in interval notation.

STEP 3

Graph the function $y = x^{1/3}$ . This is the cube root function, which passes through the origin and has points such as $(-1, -1)$ , $(0, 0)$ , and $(1, 1)$ .

STEP 4

Graph the function $y = x$ . This is the identity function, which is a straight line passing through the origin with a slope of 1.

STEP 5

Find the points of intersection of the graphs $y = x^{1/3}$ and $y = x$ . These occur where $x^{1/3} = x$ .
To find the points of intersection analytically, solve $x^{1/3} = x$ : $x^{1/3} = x$ Cubing both sides: $x = x^3$ Rearranging gives: $x^3 - x = 0$ Factoring out $x$ : $x(x^2 - 1) = 0$ Further factoring: $x(x - 1)(x + 1) = 0$ Thus, the solutions are: $x = 0, x = 1, x = -1$ These points are where the graphs intersect.

STEP 6

Determine the intervals where $x^{1/3} < x$ based on the graphs.
From the points of intersection:
1. For $x < -1$ , $x^{1/3}$ is greater than $x$ .
2. For $-1 < x < 0$ , $x^{1/3}$ is less than $x$ .
3. For $0 < x < 1$ , $x^{1/3}$ is less than $x$ .
4. For $x > 1$ , $x^{1/3}$ is greater than $x$ .
Therefore, the solution intervals where $x^{1/3} < x$ are $(-1, 0) \cup (0, 1)$ .

STEP 7

Write the final solution in interval notation:
$(-1, 0) \cup (0, 1)$

Was this helpful?

Studdy Solution

STEP 1

1. We need to solve the inequality $x^{1/3} < x$ .
2. We will use graphical methods to find the solutions.
3. The solutions should be stated in interval notation and rounded to two decimal places.

STEP 2

1. Graph the function $y = x^{1/3}$ .
2. Graph the function $y = x$ .
3. Find the points of intersection of the graphs.
4. Determine the intervals where $x^{1/3} < x$ based on the graphs.
5. Write the solution in interval notation.

STEP 3

Graph the function $y = x^{1/3}$ . This is the cube root function, which passes through the origin and has points such as $(-1, -1)$ , $(0, 0)$ , and $(1, 1)$ .

STEP 4

Graph the function $y = x$ . This is the identity function, which is a straight line passing through the origin with a slope of 1.

STEP 5

STEP 6

STEP 7

Write the final solution in interval notation:
$(-1, 0) \cup (0, 1)$

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

1. We need to solve the inequality x1/3<x x^{1/3} < x x1/3<x.2. We will use graphical methods to find the solutions.3. The solutions should be stated in interval notation and rounded to two decimal places.

STEP 2

1. Graph the function y=x1/3 y = x^{1/3} y=x1/3.2. Graph the function y=x y = x y=x.3. Find the points of intersection of the graphs.4. Determine the intervals where x1/3<x x^{1/3} < x x1/3<x based on the graphs.5. Write the solution in interval notation.

STEP 3

Graph the function y=x1/3 y = x^{1/3} y=x1/3. This is the cube root function, which passes through the origin and has points such as (−1,−1)(-1, -1)(−1,−1), (0,0)(0, 0)(0,0), and (1,1)(1, 1)(1,1).

STEP 4

Graph the function y=x y = x y=x. This is the identity function, which is a straight line passing through the origin with a slope of 1.

STEP 5

STEP 6

STEP 7

Write the final solution in interval notation:(−1,0)∪(0,1) (-1, 0) \cup (0, 1) (−1,0)∪(0,1)

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. We need to solve the inequality $x^{1/3} < x$ .
2. We will use graphical methods to find the solutions.
3. The solutions should be stated in interval notation and rounded to two decimal places.

1. Graph the function $y = x^{1/3}$ .
2. Graph the function $y = x$ .
3. Find the points of intersection of the graphs.
4. Determine the intervals where $x^{1/3} < x$ based on the graphs.
5. Write the solution in interval notation.

Graph the function $y = x^{1/3}$ . This is the cube root function, which passes through the origin and has points such as $(-1, -1)$ , $(0, 0)$ , and $(1, 1)$ .

Graph the function $y = x$ . This is the identity function, which is a straight line passing through the origin with a slope of 1.

Write the final solution in interval notation:
$(-1, 0) \cup (0, 1)$