**Introduction**

In this lesson, you will learn about two different types of equations: equations with infinite solutions and equations with no solution.

This video illustrates the lesson material below. Watching the video is optional.

- Special Cases Solution with Infinite Solutions and Equations with No Solution (05:35 mins) | Transcript

**Infinite Solutions**

There are three types of answers you can get when solving for a variable:

- \(x=a\): where
*a*represents all real numbers.- Example: \( x = 5.3\), \(x = -2\), \(x = 1.5\)

- \(x =\) Infinitely Many Solutions: where
*x*represents all real numbers or infinitely many solutions.- Example: \(3 = 3\), \(x =x \), \(-7 = -7\)

- \(x =\) No Solution: no solution is when the statement is false.
- Example: \(1 \neq 5\), \(0 \neq 4\), \(3 \neq -2\)

Not all equations will end with \(x =\) a specific number. Some equations may have infinitely many solutions and other equations may have no solution at all.

**Example 1: Infinite Solutions**

\begin{align*}2x + 3 &= 2x + 3 &\color{red}\small\text{Given equation}\\\\2x +3 \color{red}\mathbf{-3} &= 2x +3 \color{red}\mathbf{-3} &\color{red}\small\text{Subtract \(3\) from each side}\\\\ 2x &= 2x &\color{red}\small\text{Simplify each side}\\\\ \frac{2x}{\color{red}\mathbf{2}} &=\frac{2x}{\color{red}\mathbf{2}} &\color{red}\small\text{Divide each side by \(2\)}\\\\x &= x &\color{red}\small\text{Simplify each side}\\\\\end{align*}

This answer will always be true, no matter what value \(x\) might be. \(x\) can be any real number, and this would still be true; therefore, this equation has **infinite solutions**.

**Example 2: Infinite Solutions**

Simplify Example 1 another way:

\begin{align*}2x + 3 &= 2x + 3 &\color{red}\small\text{Given equation}\\\\ 2x +3 \color{red}\mathbf{-2x} &= 2x +3 \color{red}\mathbf{-2x} &\color{red}\small\text{Subtract \(2x\) from each side}\\\\ 3 &= 3 &\color{red}\small\text{Simplify each side}\\\end{align*}

When you solve an equation and get the same answer on each side of the equals sign, such as \(3\) equals \(3\), or \(x\) equals \(x\), all real numbers are the solution, or the equation has **infinite solutions**.

**Example 3: No Solutions**

\begin{align*}2x + 1 &= 2x -5 &\color{red}\small\text{Given equation}\\\\2x +1 \color{red}\mathbf{-2x} &= 2x -5 \color{red}\mathbf{-2x} &\color{red}\small\text{Subtract 2x from both sides}\\\\1 &= -5 &\color{red}\small\text{Simplify each side} \end{align*}

The two \(x\)'s from each side will cancel out, leaving \(1=-5\). This is a false statement. 1 is not equal to -5. Therefore, no matter what value you put into this equation for \(x\), it will never be true. This equation has **no solutions**.

**Example 4: No Solutions**

Simplify Example 3 another way:

\begin{align*}2x + 1 &= 2x - 5 &\color{red}\small\text{Given equation}\\\\2x + 1 \color{red}\mathbf{-1} &= 2x - 5 \color{red}\mathbf{-1} &\color{red}\small\text{Subtract 1 from each side}\\\\2x &= 2x -6 &\color{red}\small\text{Simplify each side}\\\\ 2x\color{red}\mathbf{-2x} &=2x - 6\color{red}\mathbf{-2x} &\color{red}\small\text{Subtract 2x from each side}\\\\ 0&= -6 &\color{red}\small\text{Simplify each side}\\\\\end{align*}

Again, the two \(x\)'s cancel each other out, leaving \(0=6\), which is not true. When you solve an equation and come up with a false statement like this one, there is **no solution**.

**Things to Remember**

- Anytime you solve an equation and get the same result on each side of the equal sign, or a true statement, the problem has infinite solutions and all real numbers are the solution.
- Anytime you solve an equation and find a false statement at the end, it means there are no solutions.

### Practice Problems

**Identify whether there is one solution, infinitely many solutions, or no solution:**

- \(-9{\text{M}} {-} 4 = -9{\text{M}} - 4\) (Solution
- \(9 + 8{\text{T}} = 13{\text{T}} + 2\) (Solution
- \(-4 + 2{\text{b}} = 2{\text{b}} - 9\) (Solution
- \(-7 + 7{\text{b}} + 18 = 3{\text{b}} + 3 - 4{\text{b}}\) (Solution
- \(2{\text{x}} + 5 + {\text{x}} = -1 + 3{\text{x}} + 6\) (Solution
- \(2(3{\text{X}} + 4) = 6{\text{X}} + 7\) (Video Solution
- \(-4(4{\text{M}} {-} 3) = -16{\text{M}} + 12\) (Video Solution

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