Problem 39 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ 4(2 z+1)<4 $$

Step-by-Step Solution

Answer

The solution is \( \{ z \,|\, z < 0 \} \).

Step 1: Distribute the Constant

First, we distribute the 4 on the left side of the inequality: \[4(2z + 1) = 8z + 4\]The inequality becomes:\[8z + 4 < 4\]

Step 2: Isolate the Variable Term

Next, subtract 4 from both sides of the inequality to isolate the term with the variable:\[8z + 4 - 4 < 4 - 4\]This simplifies to:\[8z < 0\]

Step 3: Solve for the Variable

Divide both sides of the inequality by 8 to solve for \( z \):\[\frac{8z}{8} < \frac{0}{8}\]This simplifies to:\[z < 0\]

Step 4: Write the Solution in Set Notation

Finally, express the solution in set notation. The inequality \( z < 0 \) means that all real numbers less than 0 satisfy the inequality. Therefore, the solution in set notation is:\[\{ z \,|\, z < 0 \}\]

Key Concepts

Solution Set Notation

When we solve inequalities, it's important not only to find the solution but also to express it clearly. Solution set notation is a way of representing all the possible solutions that satisfy the inequality. It uses a set format to express this range of solutions concisely.

Think of solution set notation like a container for all the numbers that make the inequality true. For example, if we have an inequality solution of \( z < 0 \), we say that the solution set includes all \( z \) values that are less than zero. To write this in solution set notation, we use the format:

• \( \{z \,|\, z < 0\} \)

This can be read as "the set of all \( z \) such that \( z \) is less than zero."

Using solution set notation gives us a standard way to communicate the solutions of inequalities, especially helpful when dealing with complex problems. It ensures clarity and avoids confusion.

Distributing Constants

Distributing constants is a common step in solving equations and inequalities. It involves taking a constant outside a parenthesis and multiplying it with each term inside. This step ensures the expression is simplified correctly before proceeding further.

In our example, we started with the inequality \( 4(2z + 1) < 4 \). The constant 4 needs to be distributed to both terms inside the parenthesis:

• Multiply 4 times \( 2z \), giving \( 8z \)
• Multiply 4 times 1, giving 4

This simplifies our expression to \( 8z + 4 < 4 \).

Distributing constants can sometimes be overlooked, but it's crucial. It ensures each term within the parenthesis is being appropriately accounted, making the inequality or equation easier to solve. By properly distributing constants, you lay the groundwork for correctly isolating variables, which is the next step.

Isolating Variable Terms

Isolating the variable terms is a vital step when solving inequalities. The main goal is to get the variable by itself on one side of the inequality, so you can clearly solve for it.

In our example, after distributing the constant, we had \( 8z + 4 < 4 \). The first step in isolating the variable is to move the constant term to the opposite side. We achieve this by subtracting 4 from both sides:

• \( 8z + 4 - 4 < 4 - 4 \)

This simplifies to \( 8z < 0 \). Now, the term with the variable is isolated.

Isolating the variable terms simplifies the inequality and sets up the next step, which is solving for the variable. It is an essential procedure to effectively handle more complex inequalities, allowing for straightforward comparison of variable values.

Inequality Solutions

Solving an inequality means finding all values that make the inequality true. Unlike equations, inequalities can have a range of possible solutions rather than a single value.

Once you've isolated the variable, as shown with \( 8z < 0 \), the next step is to solve for the variable to understand the solution range. This involves performing operations that maintain the inequality's truth. In our example, we divided both sides by 8:

• \( \frac{8z}{8} < \frac{0}{8} \)

Resulting in the simplified form \( z < 0 \), this gives us the range of solutions.

It's important to remember that dividing or multiplying by a negative number in inequalities reverses the inequality sign, but in this case, we divided by a positive number, so the inequality direction remains the same.

Understanding inequality solutions allow you to capture all possible values satisfying the condition. This foundation aids in tackling more advanced problems, ensuring you grasp the range of values as answers, rather than looking for a single solution.