Question

Complete each table by evaluating the expression using the given values for the variable. $$\begin{array}{|r|l|} \hline y & |y+2| \\ \hline-4 & \\ \hline-3 & \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline \end{array}$$

Short answer

Question: Evaluate the expression \(|y+2|\) for the given values of \(y\): -4, -3, -2, -1, and 0. Answer: For the given values of \(y\), the expression \(|y+2|\) evaluates as follows: 1. When \(y=-4\), \(|y+2| = |(-4)+2| = |-2| = 2\). 2. When \(y=-3\), \(|y+2| = |(-3)+2| = |-1| = 1\). 3. When \(y=-2\), \(|y+2| = |(-2)+2| = |0| = 0\). 4. When \(y=-1\), \(|y+2| = |(-1)+2| = |1| = 1\). 5. When \(y=0\), \(|y+2| = |(0)+2| = |2| = 2\).

Step-by-step solution

Understand the concept of absolute value

The absolute value \(|x|\) of a real number \(x\) is its distance from 0 on the number line. In other words, \(|x|\) is the value of \(x\) when its sign is removed. There are two cases: 1. If \(x >= 0\), then the absolute value of \(x\) is \(x\). Therefore, \(|x| = x\). 2. If \(x < 0\), then the absolute value of \(x\) is \(-x\). Therefore, \(|x| = -x\).

Evaluate the expression for each value of the variable

Begin by calculating the expression \(|y+2|\) for each given value of \(y\). The table below is updated with the results. $$\begin{array}{|r|l|} \hline y & |y+2| \\\ \hline-4 & |-4+2| = |-2| = 2 \\\ \hline-3 & |-3+2| = |-1| = 1 \\\ \hline-2 & |-2+2| = |0| = 0 \\\ \hline-1 & |-1+2| = |1| = 1 \\\ \hline 0 & |0+2| = |2| = 2 \\\ \hline \end{array}$$

Key concepts

Evaluating Expressions

Evaluating mathematical expressions is a fundamental skill in algebra that involves finding the value of an expression by substituting variables with numbers and following the order of operations. Think of an expression as a recipe. In the same way you must follow specific steps to correctly bake a cake, you must substitute variables and carry out operations in a particular order to correctly evaluate an expression. For instance, with the expression \( |y+2| \) from the exercise, you first substitute the values for \( y \) and then carry out the operation of taking the absolute value.

It's like solving a small puzzle: first, you decode what the letters and symbols in the expression represent, then you carefully replace these letters with the provided numbers, and finally, you apply the mathematical operations such as addition, subtraction, and in this case, finding the absolute value. This step-by-step approach ensures that you can correctly evaluate expressions from the simplest to the most complex ones.

Number Line

The number line is a straight line that graphically shows numbers as points that are spaced according to their value. Each point on the line corresponds to a number, and the distance between these points reflects the difference in these numbers' values. A number line can be a powerful tool to understand the concept of absolute value, as it serves as a visual representation of the distance of a number from zero.

When looking at the number line, positive numbers are typically to the right of zero, while negative numbers are to the left. The idea is that no matter which direction you move from zero, the distance is always positive. Therefore, the absolute value of a number, whether positive or negative, is represented by the same distance from zero on the number line. This is why \( |-4+2| \) equals \( 2 \)—it's the distance you'd measure from zero to the point representing \( -2 \) on the line, which is two units to the left of zero.

Real Numbers

Real numbers encompass all the numbers that can be found on the number line, including both positive and negative numbers, as well as zero. They include rational numbers—such as fractions and integers—and irrational numbers, which cannot be expressed as a simple fraction. The absolute value function applies to all real numbers, making it a universal concept in mathematics.

The real numbers are used not just in pure mathematics but in everyday calculations involving measuring, counting, and specifying quantities. They constitute the set of possible solutions to a wide variety of real-world problems and serve as the building blocks for more complex mathematical concepts. In our table from the exercise, all the values for \( y \) and their corresponding absolute values \( |y+2| \) are real numbers, demonstrating how real numbers and their properties are ever-present in mathematical expressions and calculations.