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

  1. by trying
  2. 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 rulesyou 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 charactersso 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 charactersif 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

  1. Isolate variables using the transformation rules
  2. 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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