Three cases can come up as we are solving linear equations – equations with one solution, equations with no solutions, and equations with infinite solutions. We have already seen where an equation has one solution. Let’s explore equations that do not have any solutions and some that have an infinite number of solutions.

Equations with No Solutions

In the example above, a solution was not obtained. Using the properties of equality to isolate the variable resulted instead in the false statement [latex]4=5[/latex]. Certainly, [latex]4[/latex] is not equal to [latex]5[/latex].

Note that in the second line of the solution above, the statement [latex]2x+4=2x+5[/latex] was obtained after combining like terms on both sides. If we examine that statement carefully, we can see that it was false even before we attempted to solve it. It would not be possible for the quantity [latex]2x[/latex] with [latex]4[/latex] added to it to be equal to the same quantity [latex]2x[/latex] with [latex]5[/latex] added to it. The two sides of the equation do not balance. Since there is no value of [latex]x[/latex] that will ever make this a true statement, we say that the equation has no solution.

Be careful that you do not confuse the solution [latex]x=0[/latex] with no solution. The solution [latex]x=0[/latex] means that the value [latex]0[/latex] satisfies the equation, so there is a solution. To say that a statement has no solution means that there is no value of the variable, not even [latex]0[/latex], which would satisfy the equation (that is, make the original statement true).

As we are solving absolute value equations, it is important to be aware of special cases. An absolute value is defined as the distance of a number from [latex]0[/latex] on a number line, so the absolute value of a number must be a positive. When an absolute value expression is given to be equal to a negative number, we say the equation has no solution ([latex]DNE[/latex], for short). Notice how this happens in the next example.

Equations with Many Solutions

You have seen that if an equation has no solution, you end up with a false statement instead of a value for [latex]x[/latex]. It is possible to have an equation where any value for [latex]x[/latex] will provide a solution to the equation.

In the example below, notice how combining the terms [latex]5x[/latex] and [latex]-4x[/latex] on the left leaves us with an equation with exactly the same terms on both sides of the equal sign.

When solving, the true statement “[latex]3=3[/latex]” was obtained. When solving an equation reveals a true statement like this, it means that the solution to the equation is all real numbers, that is, there are infinitely many solutions. Try substituting [latex]x=0[/latex] into the original equation—you will get a true statement! Try [latex]x=-\dfrac{3}{4}[/latex]. It will also satisfy the equation. In fact, any real value of [latex]x[/latex] will make the original statement true.