Question

Let a, b, and c be real numbers such that$\mathbf{a}\mathbf{>}\mathbf{0}$, $\mathbf{b}\mathbf{<}\mathbf{0}$and$\mathbf{c}\mathbf{<}\mathbf{0}$. Find the sign of each expression.

$\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{-}\mathbf{b}$

$\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{a}\mathbf{+}\mathbf{bc}$

$\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{c}\mathbf{-}\mathbf{a}$

$\mathbf{\left(}\mathbf{d}\mathbf{\right)}\mathbf{a}{\mathbf{b}}^2$

Short answer

(a) Sign of $-b$is positive

(b) Sign of$a+bc$ is positive

(c) Sign of$c-a$ is negative

(d) Sign of$a{b}^2$ is positive

Step-by-step solution

. Apply the properties of negatives

If a is a real number, then the absolute value of a is given by:

$\begin{array}{l}-(-a)=a\\ (-a)b=a(-b)=-\left(ab\right)\\ (-a)(-b)=ab\\ -(a+b)=-a-b\end{array}$

where

$a>0,b>0$

. Analyze the given variables

Given $a>0$,$b<0$and$c<0$

So, $a>0,-b>0,-c>0$

. Analyze the given expression

Given -b

Since

$\begin{array}{l}b<0\\ \Rightarrow -b>\mathrm{0........}\left(1\right)\end{array}$

. Sign of given expression

From (1), sign of is-b positive.

. Apply the properties of negatives

If a is a real number, then the absolute value of a is given by:

$\begin{array}{l}-(-a)=a\\ (-a)b=a(-b)=-\left(ab\right)\\ (-a)(-b)=ab\\ -(a+b)=-a-b\end{array}$

where

$a>0,b>0$

. Analyze the given variables

Given$a>0$ ,$b<0$and$c<0$

So, $a>0,-b>0,-c>0$

. Analyze the given expression

Given bc

Since$-b>0,-c>0$

Let

$\begin{array}{l}-b=b\text{'}>0\\ -c=c\text{'}>0\\ a>0\end{array}\}\mathrm{...............}\left(1\right)$

We can write given expression as

$\begin{array}{l}bc=[-(-b\left)\right][-(-c\left)\right]\\ \Rightarrow bc=(-b\text{'})(-c\text{'})\\ \Rightarrow bc=b\text{'}c\text{'}\mathrm{................}\left(2\right)\end{array}$

. Sign of given expression

From (1) and (2),

$\begin{array}{l}bc=b\text{'}c\text{'}>0\text{and}a>0\\ \Rightarrow a+bc>0\end{array}$

So, sign of$a+bc$ is positive

. Apply the properties of negatives

If a is a real number, then the absolute value of a is given by:

$\begin{array}{l}-(-a)=a\\ (-a)b=a(-b)=-\left(ab\right)\\ (-a)(-b)=ab\\ -(a+b)=-a-b\\ -(a-b)=b-a\\ a-b=a+(-b)\end{array}$

where

$a>0,b>0$

. Analyze the given variables

Given$a>0$ ,$b<0$and$c<0$

So, $a>0,-b>0,-c>0$

. Analyze the given expression

Given$c-a$

Let

$\begin{array}{l}-c=c\text{'}>0\\ a>0\end{array}\}\mathrm{...............}\left(1\right)$

We can write given expression as

$\begin{array}{l}c-a=-(a-c)\\ \Rightarrow c-a=-(a+(-c\left)\right)\\ \Rightarrow c-a=-(a+c\text{'})\mathrm{................}\left(2\right)\end{array}$

. Sign of given expression

From (1) and (2),

$\begin{array}{l}a+c\text{'}>0\\ \Rightarrow -(a+c\text{'})<0\\ \Rightarrow c-a=-(a+c\text{'})<0\end{array}$

So, sign of is negative

. Apply the properties of negatives

If a is a real number, then the absolute value of a is given by:

$\begin{array}{l}-(-a)=a\\ (-a)b=a(-b)=-\left(ab\right)\\ (-a)(-b)=ab\\ -(a+b)=-a-b\\ a^2>\mathrm{0..............}\text{foranyvalueof}a\end{array}$

where$a>0,b>0$

. Analyze the given variables

Given $a>0$$b<0$,and$c<0$

So, $a>0,-b>0,-c>0$

. Analyze the given expression

Given$a{b}^2$

$\begin{array}{l}{b}^2>0\\ a>0\end{array}\}\mathrm{...............}\left(1\right)$

. Sign of given expression

From (1),

$a{b}^2>0$

So, sign of$a{b}^2$ is positive