Problem 34 - A Graphical Approach to College Algebra

Question

Use the methods of Examples 1 and 3 to solve the rational equation and associated inequalities given.Then, support your answer by using the \(x\) -intercept method with a calculator graph in the suggested window. (a) \(\frac{x-6}{x+2}=-1\) (b) \(\frac{x-6}{x+2}<-1\) (c) \(\frac{x-6}{x+2}>-1\) Window: \([-10,10]\) by \([-10,10]\)

Step-by-Step Solution

Answer

(a) \(x = 2\). (b) \(x < -2\). (c) \(x > 2\).

Step 1: Solve the Rational Equation

To solve \(\frac{x-6}{x+2} = -1\), start by multiplying both sides by \((x + 2)\) to eliminate the fraction. This gives us:\[x - 6 = -1(x + 2)\]Expand the equation:\[x - 6 = -x - 2\]Add \(x\) to both sides:\[2x - 6 = -2\]Add 6 to both sides:\[2x = 4\]Divide by 2:\[x = 2\]The solution to the equation is \(x = 2\). Since \(x + 2 eq 0\) for \(x = 2\), it is a valid solution.

Step 2: Solve the Inequality \(\frac{x-6}{x+2} < -1\)

Solve the inequality \(\frac{x-6}{x+2} < -1\) by using the same approach as in Step 1:Multiply both sides by \((x+2)\):\[x - 6 < -1(x + 2)\]Expand:\[x - 6 < -x -2\]Add \(x\) to both sides:\[2x - 6 < -2\]Add 6 to both sides:\[2x < 4\]Divide by 2:\[x < 2\]Additionally, consider the restriction \(x + 2 eq 0\), so \(x eq -2\). The solution to the inequality is therefore \(-\infty < x < -2\) or \(x < 2\).

Step 3: Solve the Inequality \(\frac{x-6}{x+2} > -1\)

Solve the inequality \(\frac{x-6}{x+2} > -1\):Multiply both sides by \((x+2)\):\[x - 6 > -1(x + 2)\]This simplifies to:\[x - 6 > -x - 2\]Add \(x\) to both sides:\[2x - 6 > -2\]Add 6 to both sides to isolate \(2x\):\[2x > 4\]Divide by 2:\[x > 2\]Considering \(x + 2 eq 0\) implies \(x eq -2\). So the solution is \(x > 2\).

Step 4: Verify with x-Intercept Method and Graphing

Graph the function \(y = \frac{x-6}{x+2} + 1\) and find where it intersects the x-axis in the window \([-10, 10]\) by \([-10, 10]\).- For \(\frac{x-6}{x+2} = -1\), verify that the graph crosses the x-axis at \(x = 2\).- For \(\frac{x-6}{x+2} < -1\), verify the portion of the curve below the x-axis occurs for \(x < 2\) except where \(x = -2\).- For \(\frac{x-6}{x+2} > -1\), verify the portion of the curve above the x-axis starts from \(x > 2\). Confirm these solutions using a calculator graph to support the algebraic solutions.

Key Concepts

Inequalities

Inequalities are statements that describe the relative size or order of two values. They show how one quantity relates to another by using symbols such as \(<\), \(>\), \(\leq\), or \(\geq\). In the context of rational equations, we deal with inequalities like \(\frac{x-6}{x+2}<-1\) and \(\frac{x-6}{x+2}>-1\), which compare rational expressions to a constant. Solving these inequalities involves determining the values of \(x\) for which the inequality holds.

• To solve these, you first clear the fraction by multiplying both sides by the denominator (making sure it doesn’t equal zero).
• Focus on simplifying the inequality to isolate \(x\).
• Remember the critical values that make the denominator zero, as these are excluded from the solution set.

By understanding how these inequalities work, you can find the values of \(x\) that satisfy the conditions either less than or greater than the specified constant.

Graphing

Graphing is a visual approach to understanding equations and inequalities. When dealing with rational equations, plotting a graph provides a clear picture of where certain values lie. For example, by graphing \(y = \frac{x-6}{x+2} + 1\) over the specified window, you can visually confirm the solutions reached algebraically.

• Plotting helps identify the x-intercepts, where the graph crosses the x-axis, these are important for verifying solutions.
• Check the regions above and below the x-axis. The portions that satisfy the inequalities will show as parts of the curve that lie above or below the axis.
• Visual examination helps in understanding the behavior of the function especially near points where it is undefined, like when the denominator is zero.

Graphing acts as a supportive method to algebraic solutions, providing a tangible view of where the conditions of inequalities are met.

Algebraic Solutions

Finding algebraic solutions involves manipulating equations to find the values of variables. For rational equations like \(\frac{x-6}{x+2} = -1\), the goal is to isolate \(x\) by eliminating the fraction. Here's how you achieve this:

• Multiply through by the denominator \((x + 2)\) to dispose of the fraction.
• Simplify and reorder terms to get all instances of \(x\) on one side.
• Solve the resulting equation by performing basic operations such as addition, subtraction, multiplication, or division.

Always check if the solution makes the denominator zero - if it does, the solution is not valid due to division by zero.
Rational equations often require consideration of these restrictions, so it's key to ensure your solution is meaningful within the constraints of the problem.