### Learning Outcomes

- Solve equations that have one solution, no solution, or an infinite number of solutions

There are three cases that can come up as we are solving linear equations. We have already seen one, where an equation has one solution. Sometimes we come across equations that don’t have any solutions, and even some that have an infinite number of solutions. The case where an equation has no solution is illustrated in the next examples.

## Equations with no solutions

### Example

Solve for [latex]x.[/latex] [latex]12+2x–8=7x+5–5x[/latex]

This is *not* a solution! You did *not* find a value for [latex]x[/latex]. Solving for [latex]x[/latex] the way you know how, you arrive at the false statement [latex]4=5[/latex]. Surely [latex]4[/latex] cannot be equal to [latex]5[/latex]!

This may make sense when you consider the second line in the solution where like terms were combined. If you multiply a number by [latex]2[/latex] and add [latex]4[/latex], you would never get the same answer as when you multiply that same number by [latex]2[/latex] and add [latex]5[/latex]. Since there is no value of [latex]x[/latex] that will ever make this a true statement, the solution to the equation above is *“no solution.”*

### Try It

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. “No solution” means that there is no value, not even [latex]0[/latex], which would satisfy the equation.

Also, be careful not to make the mistake of thinking that the equation [latex]4=5[/latex] means that [latex]4[/latex] and [latex]5[/latex] are values for [latex]x[/latex] that are solutions. If you substitute these values into the original equation, you’ll see that they do not satisfy the equation. This is because there is truly *no solution*—there are no values for [latex]x[/latex] that will make the equation [latex]12+2x–8=7x+5–5x[/latex] true.

### Think About It

Try solving these equations. How many steps do you need to take before you can tell whether the equation has no solution or one solution?

a) Solve [latex]8y=3(y+4)+y[/latex]

Use the textbox below to record how many steps you think it will take before you can tell whether there is no solution or one solution.

b) Solve [latex]2\left(3x-5\right)-4x=2x+7[/latex]

Use the textbox below to record how many steps you think it will take before you can tell whether there is no solution or one solution.

## Algebraic Equations with an Infinite Number of 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.

### Example

Solve for [latex]x[/latex]. [latex]5x+3–4x=3+x[/latex]

You arrive at the true statement “[latex]3=3[/latex].” When you end up with a true statement like this, it means that the solution to the equation is “all real numbers.” Try substituting [latex]x=0[/latex] into the original equation—you will get a true statement! Try [latex]x=-\frac{3}{4}[/latex], and it will also check to be a true statement!

This equation happens to have an infinite number of solutions. Any value for [latex]x[/latex] that you can think of will make this equation true. When you think about the context of the problem, this makes sense—the equation [latex]x+3=3+x[/latex] means “some number plus [latex]3[/latex] is equal to [latex]3[/latex] plus that same number.” We know that this is always true—it’s the commutative property of addition!

### Try It

In the following video, we show more examples of attempting to solve a linear equation with either no solution or many solutions.

### Example

Solve for [latex]x[/latex]. [latex]3\left(2x-5\right)=6x-15[/latex]

In this video, we show more examples of solving linear equations with either no solutions or many solutions.

In the following video, we show more examples of solving linear equations with parentheses that have either no solution or many solutions.

## Absolute value equations with no solutions

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

### Example

Solve for [latex]x[/latex]. [latex]7+\left|2x-5\right|=4[/latex]

### Example

Solve for [latex]x[/latex]. [latex]-\frac{1}{2}\left|x+3\right|=6[/latex]

In this last video, we show more examples of absolute value equations that have no solutions.

## Summary

We have seen that solutions to equations can fall into three categories:

- One solution. This is when you find the only value of the variable, such as [latex]x = 5[/latex].
- No solution, DNE (does not exist). This is when a false statement appears, like [latex]4 = 7[/latex].
- Many solutions, also called infinitely many solutions or All Real Numbers. This is when a true statement appears, like [latex]x + 3 = x + 3[/latex].

And sometimes, we don’t need to do much algebra to see what the outcome will be.