Before we dive into solving inequalities, let's review the definition and properties of inequalities. An inequality is a mathematical statement that compares two quantities using a symbol of inequality such as <, >, ≤, or ≥. The symbol < means "less than," > means "greater than," ≤ means "less than or equal to," and ≥ means "greater than or equal to."

Just like an equation, an inequality has solutions. However, unlike an equation, an inequality can have infinitely many solutions. For example, the inequality x < 5 has infinitely many solutions such as x = 4, x = 3, x = 2, and so on.

When solving inequalities, there are some properties that we need to keep in mind:

• If we add or subtract the same number from both sides of an inequality, the inequality remains true. For example, if x > 3, then x + 2 > 5.
• If we multiply or divide both sides of an inequality by a positive number, the inequality remains true. For example, if x < 4, then 2x < 8.
• If we multiply or divide both sides of an inequality by a negative number, the inequality changes direction. For example, if x > 2, then -x < -2.
• If we combine two inequalities with the same direction (both > or both <), we can add or subtract them. For example, if x > 3 and y > 2, then x + y > 5.
• If we combine two inequalities with opposite directions (one > and one <), we cannot add or subtract them. For example, if x > 3 and y < 2, we cannot combine them to get x + y < 5.

Understanding the properties of inequalities is essential when solving and graphing them. Make sure to keep these properties in mind as we move forward in the course.

Now that we understand what inequalities are, let's move on to solving linear inequalities with one variable. The process for solving linear inequalities is very similar to solving linear equations, with one key difference: the direction of the inequality symbol.

Let's start with an example:

2x + 3 < 9

To solve for x, we want to isolate it on one side of the inequality, just like we would with an equation. However, we also need to make sure we maintain the direction of the inequality symbol.

First, we'll subtract 3 from both sides:

2x < 6

Next, we'll divide both sides by 2:

x < 3

Our solution is x < 3. This means that any value of x that is less than 3 will make the inequality true. We can check this by plugging in a value, such as 2:

2(2) + 3 < 9

4 + 3 < 9

7 < 9

Since 7 is less than 9, our solution is correct.

Let's try another example:

5 - 3x > 7

First, we'll subtract 5 from both sides:

-3x > 2

Next, we'll divide both sides by -3. Remember, when we divide or multiply both sides of an inequality by a negative number, we need to flip the direction of the inequality symbol:

x < -2/3

Our solution is x < -2/3. This means that any value of x that is less than -2/3 will make the inequality true. We can check this by plugging in a value, such as -1:

5 - 3(-1) > 7

8 > 7

Since 8 is greater than 7, our solution is correct.

Remember, when solving inequalities, we always want to isolate the variable on one side of the inequality and maintain the direction of the inequality symbol.

Inequalities are mathematical expressions that compare two values. They are used to represent relationships between quantities that are not necessarily equal. In algebra, we use inequalities to represent the range of possible solutions for a given problem. In this section, we will focus on solving linear inequalities with two variables.

A linear inequality is an inequality that involves two variables and can be graphed as a straight line. To solve a linear inequality with two variables, we need to find the values of the variables that satisfy the inequality. This can be done by graphing the inequality and determining the shaded region that represents the solution set.

Let's take a look at an example:

2x + 3y < 12

To graph this inequality, we first need to find the equation of the line that represents it. We can do this by solving for y:

3y < -2x + 12

y < (-2/3)x + 4

Now we can graph the line y = (-2/3)x + 4:

The next step is to determine which side of the line represents the solution set. We can do this by choosing a test point that is not on the line, such as (0, 0). We substitute this point into the inequality:

2(0) + 3(0) < 12

0 < 12

This is true, which means that the region below the line is the solution set. We shade this region:

The solution set is the region below the line y = (-2/3)x + 4. We can write this in set-builder notation:

{(x, y) | y < (-2/3)x + 4}

Or interval notation:

(-∞, (-2/3)x + 4)

Now you know how to solve linear inequalities with two variables. Practice with different examples and make sure you understand the concepts before moving on to more advanced topics.

Graphing linear inequalities on the coordinate plane is an essential skill in Algebra. To graph a linear inequality, we need to follow the same steps as we did for graphing a linear equation, with one additional step.

The additional step is to determine which side of the line represents the solution set of the inequality. To do this, we choose a test point not on the line and substitute its coordinates into the inequality. If the inequality is true, the test point is in the solution set, and we shade the region containing the test point. If the inequality is false, the test point is not in the solution set, and we shade the other region.

Let's take an example of a linear inequality:

x + 2y ≤ 6

To graph this inequality, we first graph the line x + 2y = 6 by finding its intercepts:

When x = 0, 2y = 6, y = 3

When y = 0, x = 6

So, the intercepts are (0, 3) and (6, 0).

Plot these points on the coordinate plane and draw the line passing through them.

Now, choose a test point not on the line, say (0, 0), and substitute its coordinates into the inequality:

0 + 2(0) ≤ 6

0 ≤ 6

Since this is true, the test point (0, 0) is in the solution set, and we shade the region containing it. In this case, it is the region below the line.

Therefore, the graph of the inequality x + 2y ≤ 6 is:

Practice graphing linear inequalities on the coordinate plane by trying out different examples and choosing different test points.

Applying inequalities to real-world scenarios allows us to understand and solve problems that involve constraints or limitations. For example, let's say you have a budget of $500 for a school project. You need to buy materials such as paper, paints, and glue. You know that the cost of paper is $0.50 per sheet, the cost of paint is $5 per bottle, and the cost of glue is $2 per bottle. You also know that you need at least 100 sheets of paper, no more than 3 bottles of paint, and at least 2 bottles of glue. To make sure you don't exceed your budget, you can use inequalities to find the possible combinations of materials you can buy.

Let x be the number of sheets of paper, y be the number of bottles of paint, and z be the number of bottles of glue. We can write the following inequalities:

• x ≥ 100 (we need at least 100 sheets of paper)
• y ≤ 3 (we can't buy more than 3 bottles of paint)
• z ≥ 2 (we need at least 2 bottles of glue)
• 0.5x + 5y + 2z ≤ 500 (we can't spend more than $500)

The last inequality represents the total cost of the materials, which can't exceed our budget. We can simplify this inequality by multiplying each term by 10 to get rid of the decimal:

5x + 50y + 20z ≤ 5000

Now we can graph these inequalities on a coordinate plane to find the possible combinations of x, y, and z that satisfy all the constraints:

The shaded region represents the feasible region, which is the set of all points that satisfy all the inequalities. Any point inside this region corresponds to a valid combination of materials that we can buy with our budget. For example, the point (120, 2, 3) represents buying 120 sheets of paper, 2 bottles of paint, and 3 bottles of glue, which costs $490. This is a valid combination because it satisfies all the constraints and doesn't exceed our budget.

Using inequalities to solve real-world problems like this can help us make informed decisions and optimize our resources. Whether we're planning a budget, designing a product, or managing a project, inequalities can provide a powerful tool for modeling and analyzing constraints.