Problem 19 - Algebra Form and Function

Question

Table 3.1 shows values of \(z\) and the expression \(4-2 z\). For which values of \(z\) in the table is (a) \(\quad 4-2 z<2 ?\) (b) \(4-2 z>2 ?\) (c) \(4-2 z=2 ?\)

Step-by-Step Solution

Answer

Answer: To answer this question, you need to provide the given table of values for z and the corresponding values of the expression \(4-2z\). Follow the step-by-step solution provided above by comparing the given expression's values in the table with the values requested for each part (a), (b), and (c). Once you've found the values of z that satisfy each inequality/equality, present your findings as instructed in Step 5.

Step 1: Analyze the given table with \(z\) values and the expression \(4-2z\)

Look at the table provided and observe the list of values for \(z\) and their corresponding values for the given expression \(4-2z\). You may want to copy them down or have them readily accessible for the next steps.

Step 2: Find values of \(z\) for which \(4-2z<2\)

Compare each value of the given expression \(4-2z\) in the table to the number \(2\). Check if the \(4-2z\) value is less than \(2\). If it is, note down the corresponding value of \(z\) that results in the expression being less than \(2\). Write down all such values of \(z\).

Step 3: Find values of \(z\) for which \(4-2z>2\)

Using the same method as in Step 2, compare each value of the given expression \(4-2z\) in the table to the number \(2\). This time, check if the \(4-2z\) value is greater than \(2\). If it is, note down the corresponding value of \(z\) that results in the expression being greater than \(2\). Write down all such values of \(z\).

Step 4: Find values of \(z\) for which \(4-2z=2\)

Finally, compare each value of the given expression \(4-2z\) in the table to the number \(2\). Check if the \(4-2z\) value is equal to \(2\). If it is, note down the corresponding value of \(z\) that results in the expression being equal to \(2\). Write down all such values of \(z\).

Step 5: Present the results for each part (a), (b), and (c)

After completing Steps 2-4, you should have found the values of \(z\) that satisfy each of the inequalities/equality. Present your findings by listing the values of \(z\) for each part: (a) The values of \(z\) for which \(4-2z<2\) are: (list the values found in Step 2) (b) The values of \(z\) for which \(4-2z>2\) are: (list the values found in Step 3) (c) The values of \(z\) for which \(4-2z=2\) are: (list the values found in Step 4) This completes the exercise.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can combine numbers, variables, and operations. In the context of inequalities, they are used to represent relationships between different quantities. For instance, in our exercise, the expression \(4 - 2z\) includes constant numbers (4 and 2) and a variable \(z\). The expression shows how \(z\) is multiplied by 2 and then subtracted from 4.

Understanding each part of an algebraic expression is crucial. Here, the term \(-2z\) indicates the rate of change due to \(z\). As \(z\) increases, the whole expression decreases, which is an important feature to identify when dealing with inequalities.

Algebraic expressions form the backbone of many mathematical problems, making it important for students to familiarize themselves with this concept. Knowing how to interpret and manipulate them is key to solving equations and inequalities effectively.

Inequality Solutions

Solving inequalities involves finding which values satisfy a given condition. In our example, the inequalities: \(4 - 2z < 2\), \(4 - 2z > 2\), and \(4 - 2z = 2\) are statements that describe various relationships of the expression to the number 2.

To solve these, you need to work through each inequality step by step:

• For \(4 - 2z < 2\), rearrange the terms to isolate \(z\). Subtract 4 from both sides to get \(-2z < -2\), then divide by -2, remembering to reverse the inequality direction, resulting in \(z > 1\).
• For \(4 - 2z > 2\), similarly arrange the expression by subtracting 4, \(-2z > -2\), and divide by -2 to get \(z < 1\).
• The final \(4 - 2z = 2\) is solved by equating \(-2z = -2\), again dividing by -2 to find \(z = 1\).

Inequality solutions like these help find ranges or specific values of variables that make the statement true, providing a concrete set of answers aligning with the problem requirements.

Problem-Solving Steps

Effective problem-solving in mathematics relies on structured approaches. With this exercise, a step-by-step method guides you through the often complex process of solving inequalities. Here are simplified steps you can follow:

• Understand the problem: Begin by reading through the problem statement. Identify what is being asked and the algebraic expressions involved.
• Examine the expressions: For each part of the exercise, look at the table to match the \(z\) values with the expression \(4 - 2z\) and what condition (less than, greater than, or equal to 2) we need to solve for.
• Rearrange and solve: For each condition, rearrange the expression to isolate the variable \(z\). This requires arithmetic manipulation as outlined in the inequality solutions section.
• Verify solutions: Cross-check the solutions obtained by substituting back into the original expressions to ensure they satisfy the inequalities.
• Present findings: Once verified, clearly list out the values of \(z\) that meet the condition for each subpart (a, b, c).

By adhering to these steps, students can tackle inequality problems with clarity and accuracy, developing confidence in navigating algebraic challenges.