Question

If c is an integer, then determine whether the statement $\left|x\right|+c<0$has no solution is sometimes, always or never true.

Short answer

If c is an integer than $\left|x\right|+c<0$issometimes true.

Step-by-step solution

Step-1. Apply the concept of Variables and constant.

Variables: - In any expression or equation, the variable is an unknown quantity. It may be any alphabet or any special symbol.

Constants: - In any expression or equation, the constants are the values that are fixed.

While performing addition in an equation, the same kind of elements or like terms are added together.

Step-2. Concept of absolute value.

Absolute value: - Absolute value of any quantity simply means a positive value of that quantity.

Example: -

Consider a variable x. Here x can be both positive or negative but $\left|x\right|$will always be positive and hence the absolute value.

Step-3. Integers.

A number that is not fraction and whole in itself is called Integer. It includes positive and negative numbers along with 0.

Step-4. Solution of the given statement.

If c is an integer, then $\left|x\right|+c<0$has no solution is sometimes true.

Check the validity

$\begin{array}{l}\left|x\right|+c<0\\ \left|x\right|+c-c<-c\\ \left|x\right|<-c\end{array}$

This inequality is true when c is a negative integer and this is false when c is a positive integer.

Hence, the statement $\left|x\right|+c<0$is sometimes true.