Question

If a is a negative integer and b is a positive integer, tell whether the expression represents a positive or a negative integer. Explain your thinking.

1. $a-b$
2. $b-a$
3. $\left|a\right|+\left|b\right|$
4. $-\left|a\right|-\left|b\right|$

Short answer

1. The expression $a-b$ represents a negative integer.
2. The expression $b-a$ represents a positive integer.
3. The expression $\left|a\right|+\left|b\right|$ represents a positive integer.
4. The expression $-\left|a\right|-\left|b\right|$ represents a negative integer.

Step-by-step solution

(a)Step-1 – Apply the concept addition

Two numbers with similar signs get added and the resulting number carries the similar sign.

Two numbers with opposite signs get subtracted and the resulting number carries the sign of the larger number.

Step-2 – Given

The provided expression is $a-b$

a is a negative integer and b is a positive integer

Step-3 – Explanation

Since, a is a negative integer and b is a positive integer

Assume, $a=-1$and $b=1$

Therefore,

$\begin{array}{l}a-b\\ =\left(-1\right)-1\left[\text{Substituteassumedvaluesof}a\text{and}b\right]\\ =-1+\left(-1\right)\left[\text{Tosubtract1,additsopposite,}\left(-1\right)\right]\\ =-2\left[\text{Add}\right]\end{array}$

Hence, $a-b$represents a negative integer.

(b)Step-1 – Apply the concept addition

Two numbers with similar signs get added and the resulting number carries the similar sign.

Two numbers with opposite signs get subtracted and the resulting number carries the sign of the larger number.

Step-2 – Given

The provided expression is $b-a$

a is a negative integer and b is a positive integer

Step-3 – Explanation

Since, a is a negative integer and b is a positive integer

Assume, $a=-1$and $b=1$

Therefore,

$\begin{array}{l}b-a\\ =1-\left(-1\right)\left[\text{Substituteassumedvaluesof}a\text{and}b\right]\\ =1+1\left[\text{Tosubtract1,additsopposite,}\left(-1\right)\right]\\ =2\left[\text{Add}\right]\end{array}$

Hence, $b-a$represents a positive integer.

(c)Step-1 – Apply the concept addition

Two numbers with similar signs get added and the resulting number carries the similar sign.

Two numbers with opposite signs get subtracted and the resulting number carries the sign of the larger number.

Step-2 – Given

The provided expression is $\left|a\right|+\left|b\right|$

a is a negative integer and b is a positive integer

Step-3 – Explanation

Since, a is a negative integer and b is a positive integer

Assume, $a=-1$and $b=1$

Therefore,

$\begin{array}{l}\left|a\right|+\left|b\right|\\ =\left|-1\right|\text{+}\left|1\right|\left[\text{Substituteassumedvaluesof}a\text{and}b\right]\\ =1+1\left[\text{Absolutevalueof}\left|-1\right|\text{is1}\right]\\ =2\left[\text{Add}\right]\end{array}$

Hence, $\left|a\right|+\left|b\right|$represents a positive integer.

(d)Step-1 – Apply the concept addition

Two numbers with similar signs get added and the resulting number carries the similar sign.

Two numbers with opposite signs get subtracted and the resulting number carries the sign of the larger number.

Step-2 – Given

The provided expression is $-\left|a\right|-\left|b\right|$

a is a negative integer and b is a positive integer

Step-3 – Explanation

Since, a is a negative integer and b is a positive integer

Assume, $a=-1$and $b=1$

Therefore,

$\begin{array}{l}-\left|a\right|-\left|b\right|\\ =-\left|-1\right|-\left|1\right|\left[\text{Substituteassumedvaluesof}a\text{and}b\right]\\ =-1-1\left[\text{Absolutevalueof}\left|-1\right|\text{is1}\right]\\ =-1+(-1)\left[\text{Tosubtract1additsopposite,}\left(-1\right)\right]\\ =-2\left[\text{Add}\right]\end{array}$

Hence, $-\left|a\right|-\left|b\right|$represents a negative integer.