Chapter 1, Problem 14T

(a)

To determine

To solve: The solution of equation $x^3-9x-1=0$ through graph.

Answer to Problem 14T

Solution:

The solutions of equation $x^3-9x-1=0$ are $x=-2.94$, $-0.11$ and $x=3.05$.

Explanation of Solution

The given equation is,

$x^3-9x-1=0$

In the above equation let the left-hand side equation is,

$y=x^3-9x-1$

The graphs of equation y is shown below,

Figure (1)

The given equation is equivalent to $y=0$ and the solution is x-coordinate of intersection point of two graphs.

From Figure (1) the graph of y is equal to zero when graph intersect x-axis at $x=-2.943$, $x=-0.111$, $x=3.054$ approximately and these values of x-coordinates give the solution of given equation.

Thus, the solutions of equation $x^3-9x-1=0$ are $x=-2.94$, $-0.11$ and $x=3.05$.

(b)

To determine

To solve: The solution of inequality $x^2-1\le \left|x+1\right|$ through graph.

Answer to Problem 14T

The solution of inequality $x^2-1\le \left|x+1\right|$ is the interval $\left[-1,2\right]$.

Explanation of Solution

The given inequality is,

$x^2-1\le \left|x+1\right|$

In the above equation let left-hand equation is,

$y_1=x^2-1$

The right-hand side equation is,

$y_2=\left|x+1\right|$

The graph of equation $y_1$ and $y_2$ on the same set of axis is shown below.

Figure (2)

The given equation is equivalent to $y_1\le y_2$ and the solution of inequality is all x-values for which the graph of $y_1$ is less than or equal to $y_2$.

From Figure (2) the graph of $y_1$ and $y_2$ intersects each other at points $\left(-1,0\right)$ and $\left(2,3\right)$ in which the values of x-coordinates are $-1$, 2 and the graph of $y_1$ is on or below the graph $y_2$ when values of x lie in between $x=-1$ to $x=2$ and these values of x-coordinates give the solution of given inequality.

Thus, the solution of inequality $x^2-1\le \left|x+1\right|$ is the interval $\left[-1,2\right]$.