Problem 62 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum. $$|4+(-11)|+|-3+(-4)|$$

Step-by-Step Solution

Answer

The sum is 14.

Step 1: Calculating First Absolute Value

Firstly, calculate the expression in the first absolute value operator. It is 4+(-11), which is equal to -7. However, the absolute value of any number is always positive, so |4+(-11)| = |-7| = 7.

Step 2: Calculating Second Absolute Value

Now calculate the expression in the second absolute value operator. It is -3+(-4), which is equal to -7. The absolute value of -7 is 7, so |-3+(-4)| = |-7| = 7.

Step 3: Summing Up The Two Absolute Values

Finally, sum the two absolute values found in previous steps. The sum is 7 + 7, which is 14.

Key Concepts

Intermediate Algebra

Understanding the basics of Intermediate Algebra is crucial for mastering mathematical concepts that are used throughout various fields of study and real-life applications. This branch of mathematics builds upon the principles established in basic algebra and introduces more complex functions and operations. At this level, students encounter polynomials, factoring, exponents, radicals, and rational expressions.

One key aspect of Intermediate Algebra is learning to manipulate and simplify algebraic expressions, which often involves combining like terms, solving equations and inequalities, and understanding the properties of real numbers. This is a step up from elementary algebra where the focus is on basic operations such as addition, subtraction, multiplication, and division. A solid grasp of Intermediate Algebra is essential for progress in higher-level math classes and for developing critical thinking and problem-solving skills.

Absolute Value

The Absolute Value of a number is its distance from zero on the number line, regardless of direction. In essence, it represents the 'magnitude' of a number without considering its sign, thus it is always non-negative. For instance, the absolute value of both -5 and 5 is 5, since both are five units away from zero on the number line.

The notation for absolute value is a pair of vertical bars, for example, |x|. When dealing with operations that include absolute values, such as the exercise |4+(-11)| + |-3+(-4)|, it's crucial to remember that the result inside the vertical bars is evaluated in terms of its actual numeric value, disregarding its sign, before being used in subsequent calculations. This concept is a cornerstone in understanding how to solve equations and inequalities that involve absolute value.

Algebraic Expressions

An Algebraic Expression is a mathematical phrase that can include numbers, variables, and operation signs. It represents a value and does not include an equality or inequality sign, distinguishing it from equations. For example, 4 + x and 3y - 5 are both algebraic expressions. When working with expressions that involve absolute value, such as in the given problem, students must remember to evaluate the expressions inside the absolute value symbols before taking their magnitude.

Basic operations within algebraic expressions follow the order of operations, often remembered by the acronym PEMDAS ('Please Excuse My Dear Aunt Sally'), which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). The ability to manipulate algebraic expressions correctly is vital, as it forms the basis of solving more complex algebraic problems.