Question

CHALLENGE Determine whether each sentence is sometimes, always, or never true. Explain your reasoning.

a.$x+x=x$

b.$x+0=x$

Short answer

a.It is never true$2x\ne x$ for any value of x except for 0.

b. It is always true for all values of x.

Step-by-step solution

Part a. Step 1. Concept of equation.

A mathematical statement with an 'equal to' symbol between two algebraic expressions (with any arithmetic operations) is known as an equation.

Part a. Step 2. Write the given information.

Given relation is $x+x=x$.

Part a. Step 3. Calculation.

Consider the LHS of the given equation:

$\begin{array}{l}x+x\\ =2x\ne x\end{array}$

It is only true when $x=0$.

Hence, the given sentence is never true as$2x\ne x$ for any value of x except for 0.

Part b. Step 1. Concept of the equation.

A mathematical statement with an 'equal to' symbol between two algebraic expressions (with any arithmetic operations) is known as an equation.

Part b. Step 2. Write the given information.

Given relation is $x+0=x$.

Part b. Step 3. Calculation.

Consider the LHS of the given equation:

$\begin{array}{l}x+0\\ =x\end{array}$

Hence, the given sentence is always true for all values of x.