# What is an equation without a solution

### Special cases when solving equations

When solving an equation, these special cases can occur:

1. As a solution set are all rational numbers possible. \$\$ L = {QQ} \$\$

2. The equation is at no inserted number correct. \$\$ L = {\$\$ \$\$} \$\$

3. 0 is the solution to the equation. \$\$ L = {0} \$\$

### 1. All rational numbers are possible as a solution set. \$\$ L = {QQ} \$\$

Example: \$\$ 2 * x + 2 = 2 * x + 2 \$\$ You remove two \$\$ x \$\$ boxes. \$\$2=2\$\$

It creates a true statement in the last line of the solved equation.

You can now put any weight you want in the \$\$ x \$\$ box. Since you're doing it on both sides of the scale, the scales will get stuck in equilibrium.

Write down the solution set as follows: \$\$ L = {QQ} \$\$ is the unknown weight box. stands for 1 kg.

If you do another equivalent conversion, you get \$\$ 0 = 0 \$\$.

### 2. The equation is not correct for any number inserted. \$\$ L = {\$\$ \$\$} \$\$

Example: \$\$ 2 * x + 2 = 2 * x + 4 \$\$ You remove two \$\$ x \$\$ boxes. \$\$ 2 = 4 \$\$ That's one wrong statement.

The equation is not solvable. That is, the solution set is empty.

Write down the solution set as follows: \$\$ L = {\$\$ \$\$} \$\$

Summary of the two special cases:

Whenever, in the equation, the \$\$ x \$\$ceases to exist, has the equation

• either no Solution \$\$ L = {\$\$ \$\$} \$\$
• or infinitely many Solutions \$\$ L = {QQ} \$\$.

The scale model is crossed out because the scale out of balance hangs.

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### 3. 0 is the solution to the equation. \$\$ L = {0} \$\$

Example: \$\$ 5 * x = 7 * x \$\$ \$\$ | -7 * x \$\$

\$\$ - 2 * x = 0 \$\$ \$\$ |: (- 2) \$\$

\$\$ x = 0 \$\$

\$\$ L = {0} \$\$

If each \$\$ x \$\$ box weighs \$\$ 0 \$\$ kg, the scales are in equilibrium. This reshaping is not permitted:

\$\$ 5 x = 7 x \$\$ \$\$ |: x \$\$

\$\$5=7\$\$

Here you would assume that \$\$ x \$\$ is not \$\$ 0 \$\$, because you cannot divide by 0. The 0 is just the solution.