Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Solve each group of equations and inequalities analytically. (a) \(|-9-3 x|=6\) (b) \(|-9-3 x| \geq 6\) (c) \(|-9-3 x| \leq 6\)

Step-by-Step Solution

Answer

(a) \(x = -5\) or \(x = -1\); (b) \(x \leq -5\) or \(x \geq -1\); (c) \(-5 \leq x \leq -1\).

Step 1: Understanding Absolute Value Equation

To solve the equation \(|-9-3x|=6\), understand that the absolute value \(|A|=B\) translates to two cases: \(A = B\) or \(A = -B\). Here, \(A\) is \(-9-3x\).

Step 2: Solve Case 1: Without Negative

For the first case, where \(-9-3x = 6\), we solve the equation as follows:1. Add 9 to both sides: \(-3x = 15\).2. Divide by -3: \(x = -5\).

Step 3: Solve Case 2: With Negative

For the second case, where \(-9-3x = -6\), we solve the equation as follows:1. Add 9 to both sides: \(-3x = 3\).2. Divide by -3: \(x = -1\).

Step 4: Understanding Absolute Value Inequality (Greater Than or Equal)

To solve \(|-9-3x| \geq 6\), rewrite it as two inequalities: \(-9-3x \geq 6\) or \(-9-3x \leq -6\).

Step 5: Solve Inequality 1: Positive Part

For \(-9-3x \geq 6\):1. Add 9 to both sides to get \(-3x \geq 15\).2. Divide by -3 and reverse the inequality sign: \(x \leq -5\).

Step 6: Solve Inequality 2: Negative Part

For \(-9-3x \leq -6\):1. Add 9 to both sides to get \(-3x \leq 3\).2. Divide by -3 and reverse the inequality sign: \(x \geq -1\).

Step 7: Combine Inequalities

Combine the solutions to get the range: \(x \leq -5\) or \(x \geq -1\).

Step 8: Understanding Absolute Value Inequality (Less Than or Equal)

To solve \(|-9-3x| \leq 6\), rewrite it as two inequalities: \(-9-3x \leq 6\) and \(-9-3x \geq -6\).

Step 9: Solve Inequality 3: Less Than Part

For \(-9-3x \leq 6\):1. Add 9 to both sides to get \(-3x \leq 15\).2. Divide by -3 and reverse the inequality sign: \(x \geq -5\).

Step 10: Solve Inequality 4: More Than Negative Part

For \(-9-3x \geq -6\):1. Add 9 to both sides to get \(-3x \geq 3\).2. Divide by -3 and reverse the inequality sign: \(x \leq -1\).

Step 11: Find Intersection of Intervals

Intersect the intervals from the two inequalities to get the final solution: \(-5 \leq x \leq -1\).

Key Concepts

Understanding Inequalities

Inequalities express a range of possible values that can satisfy an equation. In mathematics, inequalities are represented using symbols like ">", "<", "≥" (greater than or equal to), and "≤" (less than or equal to). Unlike equations that have exact solutions, inequalities allow for a range of solutions.

When dealing with inequalities, such as \(-9-3x \geq 6\), it's essential to remember a few rules:

• Reversing the inequality sign: Whenever you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
• Combining solutions: When the solution sets of two inequalities are combined, they may need to be expressed as a range or multiple intervals, depending on whether it's a "less than" or "greater than" type of inequality.

Understanding these rules is crucial when dealing with absolute value inequalities, like those in the exercise. In practical terms, inequalities help us understand boundaries and limits, which is particularly useful in fields like economics and physics.

Analytical Solving Techniques

Analytically solving absolute value equations involves breaking down the problem into manageable steps. The main idea is to convert the absolute value into two possible linear equations.

Consider the equation \(|-9-3x| = 6\): this equation suggests that the expression inside the absolute value sign can be either positive or negative. Therefore, two cases arise:

• The positive case, \-9-3x = 6\: solving gives us \x = -5\.
• The negative case, \-9-3x = -6\: solving gives us \x = -1\.

Analytical solving means simplifying through algebraic manipulations without relying on numerical approximations. It's about understanding the logic behind the math. For inequalities, using analytical methods helps clarify where the solution sets overlap or diverge. These techniques are foundational in understanding more complex mathematical concepts in calculus and beyond.

Role of Precalculus in Absolute Value Problems

Precalculus lays the groundwork for calculus and involves a wide range of mathematical concepts, including inequalities and absolute values. It requires understanding not just procedural solving, but also the graphical representations and the real-world implications of equations and inequalities.

Working with absolute value inequalities, like \(-9-3x \leq 6\), gives precalculus students a platform to apply rules from algebra within a complex context. The solution \-5 \leq x \leq -1\ indicates a specific range where the inequality holds true. This range can also be visualized on a number line, helping students understand the spatial aspect of math.

• Graphical insights: Solving inequalities often involves understanding how changes to one side of an inequality affect the other.
• Mathematical modeling: Precalculus helps students build skills to model real-life scenarios using equations, expanding their analytical thinking.

Effective mastery of these concepts ensures that students are prepared for the challenges of calculus, where functions become more dynamic, and modeling becomes more intricate.