Problem 2 - Elementary Algebra

Question

For Problems \(1-10\), determine whether each numerical inequality is true or false. (Objective 1) $$ 5+6(-3)-8(-4)>17 $$

Step-by-Step Solution

Answer

The inequality is true.

Step 1: Calculate the Multiplications

To solve the inequality, start by calculating the multiplications: \(6(-3)\) and \(-8(-4)\). This simplifies to \(-18\) and \(+32\) respectively.

Step 2: Sum Up the Values

Next, substitute the simplified values back into the inequality: \(5 + (-18) + 32\). Calculate the sum, which results in: \(-18 + 5 = -13\), and then \(-13 + 32 = 19\).

Step 3: Evaluate the Inequality

Now, compare the result \(19\) to \(17\): since \(19 > 17\), the inequality is true.

Key Concepts

Understanding Numerical Inequalities

Numerical inequalities are expressions that compare two numbers or algebraic expressions using symbols like \(>\), \(<\), \(\geq\), and \(\leq\). These symbols help us understand the relationship between the numbers being compared.
In our exercise, we look at the inequality \(5 + 6(-3) - 8(-4) > 17\). The goal is to evaluate whether this statement is true or false by simplifying and solving each part of the inequality.
The problem involves arithmetic operations such as multiplication and addition, which should be carried out according to the order of operations (also known as BODMAS/BIDMAS - Brackets, Orders, Division and Multiplication, Addition and Subtraction). This order is crucial in obtaining the correct result and ensures the inequality is evaluated correctly.

Following Step-by-Step Solutions

Approaching algebra problems with step-by-step solutions is essential for understanding each part of the process. In the provided solution, each step is meticulously broken down.

• Step 1: Begin by handling multiplications, as they have precedence in the order of operations. Calculate \(6(-3)\) which results in \(-18\), and \(-8(-4)\) which results in \(+32\).
• Step 2: Substitute these results back into the inequality: \(5 -18 + 32\). Add or subtract these sequentially: first, \(5 -18 = -13\); then, \(-13 + 32 = 19\).
• Step 3: Evaluate \(19 > 17\). Since this is true, the inequality holds.

Using this methodical approach helps in tracking the flow of calculations and ensures no steps are skipped. It also underlines the usefulness of solving algebra problems one piece at a time, thus avoiding mistakes.

Problem Solving in Algebra

Problem-solving in algebra often involves understanding the rules of operations and complex equations. This process demands attention to detail and logical reasoning.
By working through examples and verifying each step, students can build confidence in handling such tasks independently. Let’s break down some integral techniques:

• Order of Operations: Always follow the correct sequence of solving parts of the equation. This ensures that you arrive at the correct value.
• Double-Check Work: After obtaining your result, reassess each step to verify calculations, reducing the chance of errors.
• Logical Reasoning: Evaluate whether your final answer makes sense within the context of the inequality being solved.

With consistent practice, the approach becomes second nature, and students become adept at solving even more complex algebraic problems. These skills are vital, not just in academic exercises but in various real-world scenarios, demonstrating the importance of algebra in diverse fields.