Chapter 1.3, Problem 1P

To determine

a.

To add:

The given numbers as indicated

Solution:

 $-6+\left(-2\right)=-8$

Explanation:

1) Concept:

(i) A real number $a$, with an absolute value, written as $\left|a\right|$$$ is always positive or zero.

(ii)  Rule for addition of real numbers:

A) Add absolute values and attach their common sign, to add two numbers with the same sign.

B) Subtract the smaller absolute value from the larger absolute value and attach the sign of the number with the larger absolute value, to add two numbers with different signs.

2) Given:

$-6+\left(-2\right)$

3) Calculation:

First we find the absolute values of the given numbers.

Hence using the rule of absolute values,

$|-6 | = 6$ and $| -2 | = 2$.

According to rule of addition of real numbers, add their absolute values.

So, $6 + 2=8$

Next, attach their common sign.

Hence, $-6+\left(-2\right)=-8$

Final statement:

$-6+\left(-2\right)=-8$

b.

To add:

The given numbers as indicated

Solution:

$5+\left(-8\right)=-3$

Explanation:

1) Concept:

(i) A real number $a$, with an absolute value, written as $\left|a\right|$$$ is always positive or zero.

(ii)  Rule for addition of real numbers:

A) Add absolute values and attach their common sign, to add two numbers with the same sign.

B) Subtract the smaller absolute value from the larger absolute value and attach the sign of the number with the larger absolute value, to add two numbers with different signs.

2) Given:

$5+\left(-8\right)$

3) Calculation:

First we find the absolute values of the given numbers.

Hence using the rule of absolute values,

$\left|5 \right|=5$ and $| -8 | = 8$.

According to rule of addition of real numbers, subtract the smaller absolute value from the larger absolute value.

$8- 5= 3$

Next, attach the sign of the number with the larger absolute value, so attach the sign of $8,$ which is negative.

Hence, $5+\left(-8\right)=-3$

Final statement:

$5+\left(-8\right)=-3$

c.

To add:

The given numbers as indicated

Solution:

$-4+9=5$

Explanation:

1) Concept:

(i) A real number $a$, with an absolute value, written as $\left|a\right|$$$ is always positive or zero.

(ii)  Rule for addition of real numbers:

A) Add absolute values and attach their common sign, to add two numbers with the same sign.

B) Subtract the smaller absolute value from the larger absolute value and attach the sign of the number with the larger absolute value, to add two numbers with different signs.

2) Given:

$-4+9$

3) Calculation:

First we find the absolute values of given numbers.

Hence, using the rule of absolute values,

$|-4 |=4$ and $\left| 9 \right| = 9$.

According to rule of addition of real numbers, subtract the smaller absolute value from the larger absolute value.

$9 - 4 = 5$

Next, attach the sign of the number with the larger absolute value, so attach the sign of $9,$ which is positive.

Hence, $-4+9=5$

Final statement:

$-4+9=5$

d.

To add:

The given numbers as indicated

Solution:

$\left(-3.2\right)+\left(-4.9\right)=-8.1$

Explanation:

1) Concept:

(i) A real number $a$, with an absolute value, written as $\left|a\right|$$$ is always positive or zero.

(ii)  Rule for addition of real numbers:

A) Add absolute values and attach their common sign, to add two numbers with the same sign.

B) Subtract the smaller absolute value from the larger absolute value and attach the sign of the number with the larger absolute value, to add two numbers with different signs.

2) Given:

$\left(-3.2\right)+\left(-4.9\right)$

3) Calculation:

First we find the absolute values of given numbers.

Hence, using the rule of absolute values,

$|-3.2 | = 3.2$ and $| -4.9 | = 4.9$.

According to rule of addition of real numbers, add their absolute values.

So, $3.2 + 4.9=8.1$

Next, attach their common sign.

Hence, $\left(-3.2\right)+\left(-4.9\right)=-8.1$

Final statement:

$\left(-3.2\right)+\left(-4.9\right)=-8.1$

e.

To add:

The given numbers as indicated

Solution:

$-\frac{3}{5}+\frac{2}{3}=\frac{1}{15}$

Explanation:

1) Concept:

(i) A real number $a$, with an absolute value, written as $\left|a\right|$$$ is always positive or zero.

(ii)  Rule for addition of real numbers:

A) Add absolute values and attach their common sign, to add two numbers with the same sign.

B) Subtract the smaller absolute value from the larger absolute value and attach the sign of the number with the larger absolute value, to add two numbers with different signs.

2) Given:

$-\frac{3}{5}+\frac{2}{3}$

3) Calculation:

First we find the absolute values of given numbers.

Hence, using the rule of absolute values,

$\left|-\frac{3}{5} \right|=\frac{3}{5}$ and $\left|\frac{2}{3} \right| =\frac{2}{3}$.

According to rule of addition of real numbers, subtract the smaller absolute value from the larger absolute value.

$\frac{2}{3} -\frac{3}{5}$

To find it, first find the least common multiple of $3$ and $5$.

LCD $(3, 5) = 15$

$\frac{2}{3} -\frac{3}{5}=\frac{2}{3}+\left(-\frac{3}{5}\right)=\left(\frac{2}{3}*\frac{5}{5}\right)+\left(-\left(\frac{3}{5}*\frac{3}{3}\right)\right)=\frac{10}{15}+\left(-\frac{9}{15}\right)=\frac{10-9}{15}=\frac{1}{15}$

Next, attach the sign of the number with the larger absolute value, so attach the sign of $\frac{2}{3}$, which is positive.

Hence,$\frac{2}{3} -\frac{3}{5}=\frac{1}{15}$

Final statement:

$-\frac{3}{5}+\frac{2}{3}=\frac{1}{15}$

f.

To add:

The given numbers as indicated

Solution:

$-\frac{5}{11}+\frac{3}{22}=-\frac{7}{22}$

Explanation:

1) Concept:

(i) A real number $a$, with an absolute value, written as $\left|a\right|$$$ is always positive or zero.

(ii)  Rule for addition of real numbers:

A) Add absolute values and attach their common sign, to add two numbers with the same sign.

B) Subtract the smaller absolute value from the larger absolute value and attach the sign of the number with the larger absolute value, to add two numbers with different signs.

2) Given:

$-\frac{5}{11}+\frac{3}{22}$

3) Calculation:

First we find the absolute values of given numbers.

Hence, using the rule of absolute values,

$\left|-\frac{5}{11} \right|=\frac{5}{11}$ and $\left|\frac{3}{22} \right| =\frac{3}{22}$.

According to rule of addition of real numbers, subtract the smaller absolute value from the larger absolute value.

$\frac{5}{11} -\frac{3}{22}$

To find it, first find least the common multiple of $3$ and $5$.

LCD $(11, 22) = 22$

$\frac{5}{11} -\frac{3}{22}=\frac{5}{11}+\left(-\frac{3}{22}\right)=\left(\frac{5}{11}*\frac{2}{2}\right)+\left(-\left(\frac{3}{22}*\frac{1}{1}\right)\right)=\frac{10}{22}+\left(-\frac{3}{22}\right)=\frac{10-3}{22}=\frac{7}{22}$

Next, attach the sign of the number with the larger absolute value, so attach the sign of $\frac{5}{11}$, which is negative.

Hence,$-\frac{5}{11}+\frac{3}{22}=-\frac{7}{22}$

Final statement:

$-\frac{5}{11}+\frac{3}{22}=-\frac{7}{22}$