Problem 54 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ |43+(-73)|+|-20| $$

Step-by-Step Solution

Answer

The result is 50.

Step 1: Solve inside the absolute value

First, solve the expression inside the absolute value symbol. Start with breaking down the expression:\[43 + (-73)\]Calculate the sum:\[43 + (-73) = 43 - 73 = -30\]

Step 2: Find the absolute value

Now find the absolute value of the result from Step 1:\[|-30| = 30\]

Step 3: Solve the second absolute value

Next, calculate the absolute value of the second number in the expression:\[|-20| = 20\]

Step 4: Add the absolute values

Finally, add the absolute values obtained from Steps 2 and 3:\[30 + 20 = 50\]

Key Concepts

Integer Addition

Integer addition involves combining two whole numbers, which can include both positive and negative numbers. The numbers used in integer addition can be understood as points on a number line.

• Positive numbers are located to the right of zero.
• Negative numbers are found to the left of zero.

To add integers with different signs, such as in the expression \(43 + (-73)\), you effectively subtract the smaller absolute value from the larger one and take the sign of the larger absolute value. In our example, since 73 has a larger absolute value than 43, the operation \(43 + (-73)\) becomes \(-30\). This scenario of adding different signed integers is a crucial concept and often involves thinking in terms of gains and losses or moving on a number line.

Step-by-Step Solution

A methodical, step-by-step solution is vital for effectively solving math problems, especially when dealing with absolute values and integer operations. Here we follow a clear sequence that leads to the right answer by breaking the problem into manageable parts. 1. **Identify Expressions Inside Absolute Values:** Start by focusing on the expressions within the absolute value bars. In the exercise, these are \(43 + (-73)\) and \(-20\).2. **Perform Calculations:** Follow through with calculations step-by-step: - Compute the result of \(43 + (-73)\) resulting in \(-30\). - Find the absolute value of \(-30\), which is 30. - Calculate the absolute value of \(-20\), which is 20.3. **Combine Results:** Add the absolute values obtained: \(30 + 20\). This well-organized strategy ensures clarity, efficiency, and prevents mistakes, crucial for mastering math competencies.

Problem Solving

Effective problem solving in mathematics involves applying logical thinking to reach a solution. In this case, the primary tool is understanding how absolute values work and how to handle integer addition. Absolute values dictate distance from zero on a number line, regardless of direction.

• First, neatly break down complex expressions as seen in \(43 + (-73)\).
• Recognize patterns and relationships, such as absolute values turning negative results positive.

Good problem solving also relies on practice and familiarity with each step in mathematical processes. Over time, you develop intuition for the most efficient methods, identified by consistently reaching accurate results through logical steps.

Educational Math Example

The given example demonstrates basic yet important mathematical concepts including absolute value and integer addition. Educational math examples like this serve multiple purposes:

• They illustrate how separate components of problems interact.
• They underscore fundamental operations, such as finding absolute values and adding integers.

By dividing the exercise into smaller tasks, as showcased in this solution, learners gain insights into systematic methods, preparing them for more advanced topics. Math examples clarify theories and demonstrate practical approaches, guiding learners through early problem-solving experiences.