Chapter 13.2, Problem 4CE

a.

To determine

To tell whether given expression

  $y_2-y_1$

positive or negative.

Answer to Problem 4CE

The expression

  $y_2-y_1$

is positive.

Explanation of Solution

Given:

Below is figure given

  

Calculation:

From the figure given , it can be said that the coordinates $\left(x_2,y_2\right)$ lies in first quadrant and $\left(x_1,y_1\right)$ lies in third quadrant. It means that $x_2,y_2$ have positive values and $x_1,y_1$ have negative values .

So , the value of expression $y_2-y_1$ is positive because the value of $y_1$ is negative which makes overall expression positive.

consider any coordinate say $\left(1,1\right)$ in first quadrant and $\left(-1,-2\right)$ in third quadrant

so ,

  $y_2-y_1=1-\left(-2\right)=3$

Therefore, the expression

  $y_2-y_1$

is positive.

b.

To determine

To tell whether given expression

  $x_2-x_1$

positive or negative.

Answer to Problem 4CE

The expression $x_2-x_1$ is positive.

Explanation of Solution

Given:

Below is figure given

  

Calculation:

From the figure given , it can be said that the coordinates $\left(x_2,y_2\right)$ lies in first quadrant and $\left(x_1,y_1\right)$ lies in third quadrant. It means that $x_2,y_2$ have positive values and $x_1,y_1$ have negative values .

So , the value of expression $x_2-x_1$ is positive because the value of $x_1$ is negative which makes overall expression positive.

consider any coordinate say $\left(1,1\right)$ in first quadrant and $\left(-1,-2\right)$ in third quadrant

so ,

  $x_2-x_1=1-\left(-1\right)=2$

Therefore, the expression

  $x_2-x_1$

is positive.

c.

To determine

To tell whether given expression

  $\frac{y_2-y_1}{x_2-x_1}$

is positive or negative.

Answer to Problem 4CE

The expression

  $\frac{y_2-y_1}{x_2-x_1}$

is positive.

Explanation of Solution

Given:

Below is figure given

  

Calculation:

From the figure given , it can be said that the coordinates $\left(x_2,y_2\right)$ lies in first quadrant and $\left(x_1,y_1\right)$ lies in third quadrant. It means that $x_2,y_2$ have positive values and $x_1,y_1$ have negative values .

So , the value of expression $x_2-x_1$ is positive because the value of $x_1$ is negative which makes overall expression positive .

And the value of expression $y_2-y_1$ is positive because the value of $y_1$ is negative which makes overall expression positive.

So, the expression

  $\frac{y_2-y_1}{x_2-x_1}$

is positive

consider any coordinate say $\left(1,1\right)$ in first quadrant and $\left(-1,-2\right)$ in third quadrant

so ,

  $x_2-x_1=1-\left(-1\right)=2$

  $y_2-y_1=1-\left(-2\right)=3$

  $\frac{y_2-y_1}{x_2-x_1}=\frac{3}{2}$

Therefore, the expression

  $\frac{y_2-y_1}{x_2-x_1}$

is positive.