# What is the formula for solving inequalities?

### What is an inequality?

A **Inequality** consists of two terms, which are separated by one of the characters $$ lt $$; $$ gt $$; $$ le $$ or $$ ge $$ are connected.

**Examples:**

$$ x + 2 gt - 8 $$

$$ x + 10 lt 20 $$

$$ - 8x - 22 + 12x le -30 $$

$$ - 3x + 1 + 5x ge -5 $$

If you put in numbers for the variables, you get **real** or **false statements**.

**Example:** $$ x-2 <4 $$

Deploy:

$$-2<4$$

$$-1<4$$ **real** statement

Deploy:

$$-2<4$$

$$0<4$$ **real** statement

Deploy:

$$-2<4$$

$$6<4$$ **wrong** statement

You see: an inequality can have several solutions.

Just like equations, you also solve inequalities

- by trying
- by forming

There is this **Comparison sign**: **$$ lt $$** Lowercase sign

$$ x <2 $$$$: $$ x is less than 2**$$ gt $$** Greater sign

$$ x> 2 $$$$: $$ x is greater than 2**$$ le $$** Less equal sign

$$ xle2 $$$$: $$ x is less than or equal to 2

$$ x $$ is at most 2**$$ ge $$** Greater or equal sign

$$ xge2 $$$$: $$ x is greater than or equal to 2

$$ x $$ is at least 2

### Solving an inequality through trial and error

**Task:**

Which natural numbers satisfy the inequality $$ 5> 7x-8 $$?

### 1st step: Inserting the trial values

Sit down **Trial values** and check whether a true statement is made. One helps **table**:**Example:**

$$ x $$ | $$ 7x-8 $$ | $$ 5 gt7x-8 $$ | Statement is |
---|---|---|---|

$$0 $$ | $$-8$$ | $$ 5 gt -8 $$ | true |

$$ 1 $$ | $$-1$$ | $$ 5 gt -1 $$ | true |

$$ 2 $$ | $$6 $$ | $$ 5> 6 $$ | not correct |

$$3$$ | $$13 $$ | $$ 5> 13 $$ | not correct |

$$4 $$ | $$20$$ | $$ 5> 20 $$ | not correct |

$$… $$ | $$…$$ | $$ …$$ | $$ …$$ |

### 2nd step: Determine the solution set L

All the numbers that, when wagered, become a **true statement** lead are one **solution** the inequality. An inequality can therefore **multiple solutions** to have.

In the example these were the numbers 0 and 1.

These numbers make up the **Solution set**

$$ L = {0; 1} $$

### In memory of

**Natural numbers:**

$$ NN = {0, 1, 2, 3, 4, 5, ...} $$ **Whole numbers:**

$$ ZZ $$ = {…, -3, -2, -1,0,1,2,3, ...}**Rational numbers:**

$$ QQ $$ = {whole numbers and fractions}

Inserting all even larger natural numbers in this example also leads to incorrect statements, since the right side of the inequality increases while the left side remains the same.

**All** Solutions to an inequality are found in the **Solution set L** summarized.

### Solving an inequality by reshaping

Like you **Inequalities** solves by trying, you know now.

It is always safest to use the entire set of solutions **to be determined arithmetically**:

You **isolate** the variable on one side of the inequality with the **Forming rules**you know from solving equations.

### Addition and subtraction rule

You can add or subtract the same number on either side of an inequality without changing the solution set.

**Example:**

$$ x - 4 lt 19 $$ $$ | + 4 $$

$$ x - 4 + 4 lt 19 + 4 $$

$$ x lt 23 $$

These are all numbers smaller than 23. You can no longer write them individually in the solution set. Then you write: $$ L = {x in QQ | xlt23} $$

say: **Set of all x from $$ QQ $$ for which the following applies: x less than 23**

### Multiplication and division rule

You can have both sides of an inequality with the same **positive** Multiply number (by the same **positive** Divide number) without changing the solution set. **Example:**

$$ 3x gt 48 |: $$$$ 3 $$

$$ 3x: 3 gt 48: 3 $$

$$ 1 * x> 16 $$

$$ x> 16 $$

$$ L = {x in QQ | xgt16} $$

These rules are the equivalence transformations.

equivalent (lat): equivalent

Depending on the task, numbers from $$ QQ $$ or $$ ZZ $$ can belong to the solution set. Then write $$ L = {x in QQ…} $$ or $$ L = {x in ZZ…} $$.

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### Beware of $$ * $$ and $$: $$ when forming

Multiply (divide) both sides of an inequality by the same **negative** Number (through the same **negative** Number), you have to **Reverse comparison characters**so that the amount of solution does not change. **Example:**

$$ - 4x according to 28 $$ $$ | $$ $$: $$

$$ - 4x: (-4) $$ $$ 28: (-4) $$ $$ rarr $$

$$ 1 * x gt - 7 $$

$$ x gt - 7 $$

$$ L = {x in QQ | xgt-7} $$

Don't forget that **To flip comparison characters**if you are with a **negative** Multiply or divide number.

### Another example

**task**:

Solve the inequality $$ - 14x + 16 <72 $$.

$$ - 14x + 16 according to 72 | -16 $$

$$ - 14x + 16 - 16 lt 72 - 16 $$

$$ - 14x lt 56 | $$ $$: $$

$$ - 14x: (-14) $$ $$ 56: (-14) $$

$$ 1⋅ x> -4 $$

$$ x> -4 $$

$$ L = {x in QQ $$ $$ | $$ $$ x> - 4} $$

### Loosening by forming

- Isolate variables using the transformation rules
- Determine the amount of solution

### An example of quadratic inequalities

**Task:**

Which natural numbers satisfy the inequality $$ x ^ 2 gt 7x-8 $$?

### 1st step: Inserting the trial values

$$ x $$ | $$ x ^ 2 $$ | $$ 7x-8 $$ | $$ x ^ 2gt7x-8 $$ | Statement? |
---|---|---|---|---|

$$0$$ | $$ 0$$ | $$-8$$ | $$ 0 gt -8 $$ | true |

$$ 1$$ | $$1$$ | $$-1$$ | $$ 1gt -1 $$ | true |

$$2$$ | $$4$$ | $$6$$ | $$ 4gt 6 $$ | not correct |

$$3$$ | $$9$$ | $$13$$ | $$ 9gt 13 $$ | not correct |

$$4$$ | $$16$$ | $$20$$ | $$ 16> 20 $$ | not correct |

$$5$$ | $$25$$ | $$27$$ | $$ 25gt 27 $$ | not correct |

$$6$$ | $$36$$ | $$34$$ | $$ 36 gt 34 $$ | true |

$$7$$ | $$49$$ | $$41$$ | $$ 49 gt 41 $$ | true |

… | … | … | … | … |

Inserting all even larger natural numbers in this example also leads to true statements, since the left-hand side of the inequality grows faster than the term on the right-hand side.

### Step 2: Determine the amount of solution

Inserting the numbers 0, 1, 6, 7 and all natural numbers greater than 7 results in a true statement.

$$ L = {0; 1; 6; 7; 8; …} $$

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