Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(\frac{x+12}{x+2} \geq 3\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x \leq 3\) and \(x \neq -2\). When graphed on the real number line, this includes all real numbers less than 3, excluding -2.

Step 1: Clear the Fraction

Transform inequality \(\frac{x+12}{x+2} \geq 3\) into an equation without fractions. Multiply both sides by \(x+2\), yielding \((x+12) \geq 3(x+2)\).

Step 2: Simplify the Inequality

Simplify the inequality to solve for x. Expand the right hand side, to get \(x+12 \geq 3x+6\). Subtract \(x\) and \(6\) on both sides, resulting in \(6 \geq 2x\). Divide both sides by \(2\) gives us the solution \(x \leq 3\).

Step 3: Identify Solution Sets

The inequality \(x \leq 3\) implies that any number less than or equal to 3 is a solution to our inequality. However remember that \(x \neq -2\), because that would make the denominator of our original inequality zero.

Step 4: Graph the Solution

On the number line, put a closed dot on 3, and an open dot on -2. Draw a line segment between -2 and 3, then extend the line to the left of -2. The open dot on -2 signifies that -2, though included in the line segment, isn't part of the solution.

Key Concepts

Inequality Graphing

Inequality graphing is a visual method to represent the solution set of an inequality on a number line. When solving inequalities like \(\frac{x+12}{x+2} \geq 3\), graphing helps to visually demonstrate where the solutions lie in relation to each other. Once the inequality is simplified, as in our example to \(x \leq 3\), we can transfer this information onto a number line.

To graph the solution set, we place a closed dot on the number 3 to indicate that 3 is included in the solution set. Since we also have to consider that \(x eq -2\), we place an open dot on -2 to show that it's not part of the solution. The line segment between -2 and 3 represents all the numbers greater than -2 and up to, and including, 3.

Finally, we extend the line segment to the left of -2, indicating that all real numbers less than -2 also satisfy the inequality. Graphing inequalities in this way provides a clear, intuitive understanding of which values are solutions.

Algebraic Fractions

Algebraic fractions, also known as rational expressions, are fractions that contain a variable, usually \(x\). In the given exercise, \(\frac{x+12}{x+2} \geq 3\), is an algebraic fraction because it has an \(x\) in both the numerator and the denominator.

Before we can solve or graph the inequality, it's important to clear the fraction. Multiplying both sides of the inequality by the denominator, \(x+2\), achieves this and results in an equation without fractions.

Understanding how to manipulate algebraic fractions by clearing them is crucial because it allows us to solve the inequality in a more straightforward algebraic form.

Real Number Line

The real number line is a fundamental concept in algebra that graphically represents all possible real numbers. It is essentially a horizontal line with a series of equally spaced marks that define a scale, usually marked at the integers. Each point on the line corresponds to a single real number, which makes it a perfect tool for showing ranges of solutions for inequalities.

In the context of solving inequalities, such as \(x \leq 3\), the real number line helps us identify which numbers make the inequality true. The area to the left or right of a point, depending on the inequality, is highlighted to indicate all the numbers that satisfy the condition. When we graph the solution set of \(x \leq 3\) on the real number line, we illustrate exactly which numbers are included.

Solution Sets

Solution sets are collections of all possible values that satisfy a given inequality. When working with an inequality like \(\frac{x+12}{x+2} \geq 3\), we must find all the values of \(x\) that make the inequality true. After solving the inequality algebraically, we get \(x \leq 3\) with the caveat that \(x eq -2\).

This means the solution set includes every real number less than or equal to 3 except -2. In this case, the solution set is represented by the interval \((-\infty, -2) \cup (-2, 3]\), indicating the union of all numbers from negative infinity to just before -2 and from just after -2 up to and including 3. This interval notation is another way to express the range of solutions besides graphing it on the real number line.