Problem 14 - College Algebra

Question

In Exercises 13-26, express each interval in terms of an inequality and graph the interval on a number line. $$(-2,4]$$

Step-by-Step Solution

Answer

The expression of the interval (-2,4] in terms of inequality is -2 < x ≤ 4.

Step 1: Translate Interval into Inequality

The interval notation (-2,4] represents all the numbers between -2 and 4, where -2 is not included (denoted by the parenthesis) and 4 is included (denoted by the square bracket). This can be translated into inequality notation as: -2 < x ≤ 4.

Step 2: Draw the Number Line

Draw a straight line and mark it as the number line. The number line should include the numbers -3 through 5 to clearly depict the space that the interval takes up. Mark the interval (-2, 4] on this line.

Step 3: Graph the Interval

On the number line, mark a hollow circle at -2 to show that -2 is not included in the interval, and a filled circle at 4 to indicate that 4 is included. Draw a line connecting -2 and 4 to represent all the values in between.

Key Concepts

Inequality Representation

Understanding inequality representation is vital for solving many algebra problems, as it helps to visualize the range of solutions to an equation or inequality. An inequality represents a comparison between two expressions and indicates that one is greater than, less than, equal to or not equal to the other. It uses symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to show these relationships. For instance, the inequality notation (-2, 4] is interpreted as 'all numbers greater than -2 and less than or equal to 4.' In algebraic terms, the corresponding inequality expression is -2 < x ≤ 4. This means all real numbers 'x' that satisfy this condition lie within the given interval. It is crucial to note the strict inequality for -2 (indicated by the < symbol), meaning that -2 is not included, while the 'less than or equal to' sign next to 4 (≤) indicates that 4 is included in the set of solutions.

Number Line Graphing

Visualizing inequalities is made simpler with the use of number line graphing. When graphing on a number line, every point corresponds to a real number and helps to show the set of all values that are solutions to an inequality. To graph the interval notation (-2, 4], start by drawing a straight horizontal line. This line symbolizes the number line which stretches endlessly in both directions but only a small section is shown for practical purposes. Mark points at regular intervals to represent numbers, including -3, -2, -1, 0, 1, 2, 3, 4, and 5 to encapsulate the interval in question. For our specific interval (-2, 4], you would draw a hollow circle at -2 to show the exclusion of this number and a filled-in circle at 4 to denote inclusion. A line drawn between these two points then represents the continuum of numbers that satisfy the inequality -2 < x ≤ 4. This intuitive method helps students understand the range of solutions without having to test individual values within the interval.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. They are the building blocks for conveying relationships in algebra and can be as simple as a single term or as complex as a lengthy polynomial. For example, the inequality we have from the interval (-2, 4] can be expressed algebraically as -2 < x ≤ 4, where 'x' represents a variable which can take on any value that satisfies the inequality. The expression itself implies a range of possible values rather than a fixed value. In learning about algebraic expressions, students must be comfortable manipulating these expressions using fundamental properties of operations to solve for variables, expand expressions, and understand complex problems. It is essential to evaluate expressions correctly, keeping in mind the order of operations and the correct use of inequality signs to ensure accurate solutions.