Öffnen – Aufgaben – Ungleichungen Klasse 7 PDF
Öffnen – Lösungen – Ungleichungen Klasse 7 PDF
Inequalities are mathematical expressions that contain variables that represent unknown quantities. The solution to an inequality is the set of values that the variable can take on for the inequality to be true.
There are three types of inequalities that you’ll encounter in math:
- greater than (>),
- less than (<), and
- greater than or equal to (>=).
You’ll also need to be familiar with the following terms:
- range: the set of values that a variable can take on,
- boundary value: a value on the edge of the range of values that a variable can take on, and
- inequality symbol: the symbol that represents the type of inequality (<, >, or >=).
How to Solve Inequalities
There are two steps to solving inequalities:
- Isolate the variable on one side of the inequality.
- Test various values for the variable to see which make the inequality true.
Let’s look at each of these steps in more detail.
Step 1: Isolate the Variable
To solve an inequality, you need to isolate the variable on one side of the inequality. This means that you need to use inverse operations to get the variable by itself on one side of the inequality.
Here are the inverse operations for each inequality symbol:
- greater than: addition and subtraction,
- less than: addition and subtraction, and
- greater than or equal to: multiplication and division.
For addition and subtraction, you want to add or subtract the same value from both sides of the inequality until the variable is by itself on one side. For multiplication and division, you want to multiply or divide both sides of the inequality by the same non-zero value until the variable is by itself on one side.
Step 2: Test Various Values
Once you’ve isolated the variable, you need to test various values for the variable to see which make the inequality true.
To do this, you need to substitute the various values into the inequality and see if the inequality is still true. If it is, then that value is part of the solution. If it’s not, then that value is not part of the solution.
There are two things to keep in mind when testing values:
- You need to test all possible values for the variable. This means that you need to test values that are less than the boundary value, the boundary value, and greater than the boundary value.
- You need to use inequality symbols when testing values. This means that you need to use a less than symbol (<) when testing values that are less than the boundary value, a greater than symbol (>) when testing values that are greater than the boundary value, and the same inequality symbol that was originally given when testing the boundary value.
Let’s look at an example to see how this works.
Example: Solve the Inequality x > 2
Let’s solve the inequality x > 2.
This inequality has the following:
- variable: x,
- boundary value: 2,
- inequality symbol: > (greater than).
To solve this inequality, we need to follow the two steps that we outlined earlier:
- Isolate the variable on one side of the inequality.
- Test various values for the variable to see which make the inequality true.
Let’s look at each of these steps in more detail.
Step 1: Isolate the Variable
The first step is to isolate the variable on one side of the inequality. To do this, we need to use inverse operations to get the variable by itself on one side of the inequality.
The inequality symbol in this inequality is > (greater than), which means that the inverse operations that we can use are addition and subtraction.
We want to add the same value to both sides of the inequality until the variable is by itself on one side. Since we’re adding 2 to both sides, we’re really just adding 0 to one side. This doesn’t change the inequality, so we’re done isolating the variable.
The inequality that we now have is:
x > 2
Step 2: Test Various Values
The second step is to test various values for the variable to see which make the inequality true.
To do this, we need to substitute the various values into the inequality and see if the inequality is still true. If it is, then that value is part of the solution. If it’s not, then that value is not part of the solution.
There are two things to keep in mind when testing values:
- You need to test all possible values for the variable. This means that you need to test values that are less than the boundary value, the boundary value, and greater than the boundary value.
- You need to use inequality symbols when testing values. This means that you need to use a less than symbol (<) when testing values that are less than the boundary value, a greater than symbol (>) when testing values that are greater than the boundary value, and the same inequality symbol that was originally given when testing the boundary value.
Let’s look at each of these points in more detail.
The first point is that we need to test all possible values for the variable. This means that we need to test values that are less than the boundary value, the boundary value, and greater than the boundary value.
The boundary value in this inequality is 2. This means that we need to test the following values:
- less than 2: 1, 0, -1, -2, -3, …
- 2: 2
- greater than 2: 3, 4, 5, 6, …
The second point is that we need to use inequality symbols when testing values. This means that we need to use a less than symbol (<) when testing values that are less than the boundary value, a greater than symbol (>) when testing values that are greater than the boundary value, and the same inequality symbol that was originally given when testing the boundary value.
The inequality symbol in this inequality is > (greater than). This means that we need to use the following inequality symbols when testing values:
- less than 2: <
- 2: >
- greater than 2: >
Let’s look at each of these values in more detail.
The first set of values that we need to test are values that are less than the boundary value. The boundary value in this inequality is 2. This means that we need to test the following values: 1, 0, -1, -2, -3, …
We need to use a less than symbol (<) when testing these values. This means that the inequalities that we need to test are:
- x < 1
- x < 0
- x < -1
- x < -2
- x < -3
We can test these values by substituting them into the original inequality and seeing if the inequality is still true. If it is, then that value is part of the solution. If it’s not, then that value is not part of the solution.
Let’s look at each of these inequalities in more detail.
x < 1
When we substitute 1 for x, we get:
1 < 1
This inequality is true, which means that 1 is part of the solution.
x < 0
When we substitute 0 for x, we get:
0 < 0
This inequality is true, which means that 0 is part of the solution.
x < -1
When we substitute -1 for x, we get:
-1 < -1
This inequality is true, which means that -1 is part of the solution.
x < -2
When we substitute -2 for x, we get:
-2 < -2
This inequality is true, which means that -2 is part of the solution.
x < -3
When we substitute -3 for x, we get:
-3 < -3
This inequality is true, which means that -3 is part of the solution.
We can continue testing values that are less than the boundary value in this way. However, we can see that all of the values that we’ve tested so far are part of the solution.
This means that we don’t need to test any more values that are less than the boundary value. We can move on to testing the boundary value and values that are greater than the boundary value.
The next set of values that we need to test is the boundary value. The boundary value in this inequality is 2. This means that we need to test the following value: 2.
We need to use the same inequality symbol that was originally given when testing the boundary value. The inequality symbol in this inequality is > (greater than). This means that the inequality that we need to test is:
x > 2
We can test this value by substituting it into the original inequality and seeing if the inequality is still true. If it is, then that value is part of the solution. If it’s not, then that value is not part of the solution.
Let’s look at this inequality in more detail.
x > 2
When we substitute 2 for x, we get:
2 > 2
This inequality is not true, which means that 2 is not part of the solution.
The last set of values that we need to test are values that are greater than the boundary value. The boundary value in this inequality is 2. This means that we need to test the following values: 3, 4, 5, 6, …
We need to use a greater than symbol (>) when testing these values. This means that the inequalities that we need to test are:
- x > 3
- x > 4
- x > 5
- x > 6
We can test these values by substituting them into the original inequality and seeing if the inequality is still true. If it is, then that value is part of the solution. If it’s not, then that value is not part of the solution.
Let’s look at each of these inequalities in more detail.
x > 3
When we substitute 3 for x, we get:
3 > 3
This inequality is not true, which means that 3 is not part of the solution.
x > 4
When we substitute 4 for x, we get:
4 > 4
This inequality is not true, which means that 4 is not part of the solution.
x > 5
When
Aufgaben mit Lösungen Ungleichungen Klasse 7
Inequalities are mathematical expressions that contain variables that represent values that can be greater than or less than other values.
There are many different types of inequalities, but the most common are linear inequalities. A linear inequality is an inequality that can be represented on a graph by a straight line.
In this lesson, we will learn how to solve linear inequalities. We will also practice solving some inequalities of our own.
Solving Linear Inequalities
To solve a linear inequality, we need to find the values of the variable that make the inequality true. We can do this by using a number line.
Let’s look at an example:
x + 2 > 5
To solve this inequality, we need to find the values of x that make the inequality true. We can do this by using a number line.
First, we’ll graph the inequality:
x + 2 > 5
We can see that the values of x that make the inequality true are all the values to the right of 3 on the number line. These are the values of x that we need to find.
To find these values, we can use a number line. We’ll start at 3 and count to the right:
3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
We can see that the values of x that make the inequality true are all the values to the right of 3 on the number line. These are the values of x that we need to find.
Now, let’s look at another example:
x – 4 < 2
To solve this inequality, we need to find the values of x that make the inequality true. We can do this by using a number line.
First, we’ll graph the inequality:
x – 4 < 2
We can see that the values of x that make the inequality true are all the values to the left of 6 on the number line. These are the values of x that we need to find.
To find these values, we can use a number line. We’ll start at 6 and count to the left:
5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12
We can see that the values of x that make the inequality true are all the values to the left of 6 on the number line. These are the values of x that we need to find.
Now, let’s try solving some inequalities of our own!
Practice
Solve the following inequalities:
x + 5 < 3
The values of x that make the inequality true are all the values to the right of -2 on the number line. These are the values of x that we need to find.
To find these values, we can use a number line. We’ll start at -2 and count to the right:
-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17
We can see that the values of x that make the inequality true are all the values to the right of -2 on the number line.
x – 3 > 4
The values of x that make the inequality true are all the values to the left of 7 on the number line. These are the values of x that we need to find.
To find these values, we can use a number line. We’ll start at 7 and count to the left:
6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12
We can see that the values of x that make the inequality true are all the values to the left of 7 on the number line.
