Problem 26 - Algebra Form and Function

Question

Table 4.9 shows values of \(x\) and the expression \(3 x+2\). For which values of \(x\) in the table is (a) \(3 x+2<8 ?\) (b) \(3 x+2>8 ?\) (c) \(3 x+2=8 ?\) Table 4.9 $$ \begin{array}{c|c|c|c|c|c} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline 3 x+2 & 2 & 5 & 8 & 11 & 14 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Question: For the given values of x in Table 4.9, determine the values of x for which the expression 3x+2 is (a) less than 8, (b) greater than 8, and (c) equal to 8. Answer: (a) The values of \(x\) for which \(3x+2<8\) are \(x=0\) and \(x=1\). (b) The values of \(x\) for which \(3x+2>8\) are \(x=3\) and \(x=4\). (c) The value of \(x\) for which \(3x+2=8\) is \(x=2\).

Step 1: Identify the given values of x and the expression 3x+2

According to Table 4.9, the given values of x are 0, 1, 2, 3, and 4. The corresponding values of the expression \(3x+2\) are 2, 5, 8, 11, and 14 respectively.

Step 2: Check which of the given values of x satisfy the condition 3x+2 < 8

Compare the given values of the expression \(3x+2\) with 8. If the value of the expression is less than 8, then it satisfies the condition. In this case, the values 2 and 5 are less than 8. These correspond to x values of 0 and 1. The values of \(x\) that satisfy this condition are \(x=0\) and \(x=1\).

Step 3: Check which of the given values of x satisfy the condition 3x+2 > 8

Compare the given values of the expression \(3x+2\) with 8. If the value of the expression is greater than 8, then it satisfies the condition. In this case, the values 11 and 14 are greater than 8. These correspond to x values of 3 and 4. The values of \(x\) that satisfy this condition are \(x=3\) and \(x=4\).

Step 4: Check which of the given values of x satisfy the condition 3x+2 = 8

Compare the given values of the expression \(3x+2\) with 8. If the value of the expression is equal to 8, then it satisfies the condition. In this case, the value 8 is equal to 8. This corresponds to an x value of 2. The value of \(x\) that satisfies this condition is \(x=2\).

Step 5: Write the final answer

(a) The values of \(x\) for which \(3x+2<8\) are \(x=0\) and \(x=1\). (b) The values of \(x\) for which \(3x+2>8\) are \(x=3\) and \(x=4\). (c) The value of \(x\) for which \(3x+2=8\) is \(x=2\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They do not include an equals sign, unlike equations.
For instance, the expression \(3x + 2\) consists of a coefficient, a variable, and a constant.

• The coefficient is the number multiplied by the variable, which here is 3.
• The variable is a symbol that may represent different values, represented by \(x\) in this case.
• The constant is a fixed value added to the term, which is 2 here.

Understanding how to manipulate and evaluate algebraic expressions is vital in solving many mathematical problems. You often substitute a variable with a specific number to simplify or clarify what the expression means.
In our original exercise, we replace \(x\) with each number in the set \( \{0, 1, 2, 3, 4\} \) to evaluate the expression \(3x + 2\). This helps us to later compare the resulting values to satisfy given conditions.

Inequality Solutions

Inequality solutions involve finding values for a variable that satisfy a condition represented by an inequality symbol. Common inequality symbols include \(<, \), \(>, \), \(\leq\), and \(\geq\).
In the exercise, you deal with three types of inequalities:

• \(3x + 2 < 8\): We want the expression \(3x + 2\) to be less than 8.
• \(3x + 2 > 8\): We want the expression \(3x + 2\) to be greater than 8.
• \(3x + 2 = 8\): We want the expression \(3x + 2\) to be exactly equal to 8.

After substituting the values of \(x\) from the given table, you check which substitutions successfully make the inequality true. For instance, when \(x = 0\) and \(x = 1\), the expression yields values 2 and 5, both of which are less than 8, satisfying \(3x + 2 < 8\). Solving inequalities enables you to determine ranges of valid values, essential in many real-world scenarios, such as budgeting or measuring physical limits.

Mathematical Equations

Mathematical equations involve expressions set equal to each other, including an equals sign. They assert that the two sides of the equals sign represent the same quantity, allowing you to solve for unknown variables.
At times, equations can be seen as specific types of inequalities, where the objective is to find where two expressions hold a precise value.
For example, in the original exercise's equation \(3x + 2 = 8\), solving an equation means finding the value of \(x\) that makes the statement true.
Here's how it works:

• Start by isolating the variable on one side.
• Subtract 2 from both sides of the equation: \(3x = 6\).
• Divide each side by 3 to solve for \(x\), yielding \(x = 2\).

Solving equations is fundamental in mathematics because it provides precise solutions to problems. Understanding and applying these principles can aid you in various fields, from physics to finance.