What value could you write in after $3 x$ that would make the equation true for: $\newline$ $1$ . no values of $x$ ? $\newline$ $2$ . all values of $x$ ? $\newline$ $3$ . just one value of $x$ ?

Full solution

Q. What value could you write in after

3 x

that would make the equation true for:

\newline

1

. no values of

x

\newline

2

. all values of

x

\newline

3

. just one value of

x

Choose Expression Greater Than $3x$ : To determine what value could be written after $3x$ to make the equation true for no values of $x$ , we need to think of an expression that would never equal $3x$ regardless of the value of $x$ . A common example of such an expression is one that is always greater or always less than any value $3x$ could take. For instance, $3x + a$ , where $a$ is a positive number, will never be equal to $3x$ because adding a positive number to $3x$ will always result in a value greater than $3x$ .
Use $3x$ for All Values: For the equation to be true for all values of $x$ , the expression after $3x$ must be something that is always equivalent to $3x$ . The simplest expression that satisfies this condition is $3x$ itself. So, if we write $3x$ after $3x$ , the equation $3x = 3x$ will be true for all values of $x$ .
Add Constant for Single Solution: To make the equation true for just one value of $x$ , we need to add a constant or another term that would make the equation solvable for a single specific value of $x$ . For example, if we write $3x + c$ , where $c$ is a constant, the equation $3x = 3x + c$ will only be true when $x$ satisfies the condition that $3x = -c$ . This will give us a single solution for $x$ .