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Understanding how to graph inequalities provides a visual way of representing a range of solutions. Let's begin with the very basics. Graphing an inequality such as \( f < 24 \) involves a number line.
For our exercise, the number less than 24 but not including 24, is represented by an open circle at 24 on the number line. An arrow extending to the left shows that all numbers to the left of 24 are solutions to the inequality because these are all the numbers less than 24. It's important to remember that the direction of the arrow always points toward the numbers that satisfy the inequality.

When to use different symbols

• Use an open circle to represent 'less than' (\( < \)) or 'greater than' (\( > \)) inequalities.
• Use a closed circle to represent 'less than or equal to' (\( \leq \)) or 'greater than or equal to' (\( \geq \)) inequalities.

Remember to always graph on a number line and use the correct symbol to visually interpret the solution set of the inequality.

To solve algebraic inequalities, it's crucial to isolate the variable on one side. This process makes it easier to see the range of possible solutions. In our exercise, we start with the inequality \( -4 + f < 20 \). The goal is to get \( f \) by itself.
Here, we add 4 to both sides, which 'moves' the -4 to the right side effectively, giving us a cleaner inequality of \( f < 24 \). This step does not change the balance of the inequality; it merely gives us a clearer view of the variable and its constraints.

General Steps for Isolation

• Add or Subtract: If there's a number added to or subtracted from your variable, do the opposite to both sides.
• Multiply or Divide: If your variable is multiplied or divided by a number, do the opposite to both sides. But watch out! If you multiply or divide by a negative number, you must flip the inequality sign.

By keeping these steps in mind, you can neatly isolate your variables and move closer to the solution.

Working with algebraic expressions is like piecing together a puzzle. In our example, the expression started as \( -4 + f < 20 \). Algebraic expressions can include variables, constants, and operations like addition, subtraction, multiplication, and division.
The role of the constants (like -4 and 20) is to frame the variable (in our case, \( f \)) in terms of what it can and can't be. When we handle these expressions, we perform arithmetic operations to simplify or rearrange the expressions, making it easier to understand the relationship between variables and constants.

Keeping Equations Balanced

• Whatever you do to one side of the inequality, you must do to the other to keep it balanced.
• Expressions can be simplified by combining like terms (terms that have the same variables to the same power).

It's this balance and simplification that make algebraic expressions both a practical tool in mathematics and a foundational concept for understanding more complex areas of math and science.