Chapter 3.3, Problem 44E

a.

To determine

To find: the value(s) of $b$ for the given situation.

Answer to Problem 44E

  $b<0$

Explanation of Solution

Given information:

The equalities are shown below has no solution.

  $\left|{\left(x-3\right)}^2-4\right|=b$

Calculation:

Plot the graph for $y=\left|{\left(x-3\right)}^2-4\right|$ as shown below.

  

Since the graphs is never negative, then it will have no solutions when $y$ or $b<0$ . Thus, for $b<0$ the equation is no solution.

b.

To determine

To find: the value(s) of $b$ for the given situation.

Answer to Problem 44E

None

Explanation of Solution

Given information:

The equalities are shown below has one solution.

  $\left|{\left(x-3\right)}^2-4\right|=b$

Calculation:

Plot the graph for $y=\left|{\left(x-3\right)}^2-4\right|$ as shown below.

  

Since the graph has at least two solutions no matter what’s the value of x because there will be no case where the equation has only one solution.

c.

To determine

To find: the value(s) of $b$ for the given situation.

Answer to Problem 44E

At $b=0$ and for all $b>4$ .

Explanation of Solution

Given information:

The equalities are shown below has two solutions.

  $\left|{\left(x-3\right)}^2-4\right|=b$

Calculation:

Plot the graph for $y=\left|{\left(x-3\right)}^2-4\right|$ as shown below.

  

There are two solutions for equation on x-axis $\left(b=0\right)$ and a point above $b=4$ . Thus, the equation has two solutions at $b=0$ and for all $b>4$ .

d.

To determine

To find: the value(s) of $b$ for the given situation.

Answer to Problem 44E

  $b=4$

Explanation of Solution

Given information:

The equalities are shown below has three solutions.

  $\left|{\left(x-3\right)}^2-4\right|=b$

Calculation:

Plot the graph for $y=\left|{\left(x-3\right)}^2-4\right|$ as shown below.

  

At an instant when $b=4$ , the graph will have three solutions.

  $\begin{array}{l}\left|{\left(x-3\right)}^2-4\right|=b\\ \left|{\left(x-3\right)}^2-4\right|=4\left[\text{Put }b=4\right]\\ {\left(x-3\right)}^2-4=4\text{ and }{\left(x-3\right)}^2-4=-4\left[\text{Solve }x\right]\\ {\left(x-3\right)}^2=4+4\text{ and }{\left(x-3\right)}^2=-4+4\\ {\left(x-3\right)}^2=8\text{ and }{\left(x-3\right)}^2=0\\ x-3=\pm \sqrt{8}\text{ and }x-3=\pm \sqrt{0}\\ x=\pm 2\sqrt{2}\text{+3 and }x=0+3\\ x=\text{3}\pm 2\sqrt{2}\text{ and }x=3\end{array}$

Thus, the equation has three solutions at $b=4$

e.

To determine

To find: the value(s) of $b$ for the given situation.

Answer to Problem 44E

  $0<b<4$

Explanation of Solution

Given information:

The equalities are shown below has three solutions.

  $\left|{\left(x-3\right)}^2-4\right|=b$

Calculation:

Plot the graph for $y=\left|{\left(x-3\right)}^2-4\right|$ as shown below.

  

There are four solutions for equation for all values of b between 0 and 4, i.e. $0<b<4$ .