Problem 5 - 5 Steps to A 5: AP Calculus AB 2021

Question

Solve the inequality \(|6-3 x|<18\) and sketch the solution on the real number line.

Step-by-Step Solution

Answer

Answer: The solution set for the inequality \(|6-3x|<18\) is the open interval \((-4, 8)\).

Step 1: Break the inequality into two cases

Consider the two cases for the absolute value: 1) \(6 - 3x \ge 0 \Rightarrow 6-3x < 18\) 2) \(6 - 3x < 0 \Rightarrow -(6-3x) < 18\)

Step 2: Solve each case

Now, let's solve each case: 1) \(6-3x < 18\) Subtract 6 from both sides: \(-3x < 12\) Divide by -3 (notice that we need to flip the inequality sign since we're dividing by a negative number): \(x > -4\) 2) \(-(6-3x) < 18\) Distribute the negative sign: \(-6+3x < 18\) Add 6 to both sides: \(3x < 24\) Divide by 3: \(x < 8\)

Step 3: Union of the solutions

Now, we'll take the union of the solutions for each case. In other words, we need to consider the values of x that satisfy either case: \(x > -4\) or \(x < 8\) This means that the solution for the inequality \(|6-3 x|<18\) is the open interval \((-4, 8)\).

Step 4: Sketch the real number line

To sketch the solution on the real number line, we need to represent the interval \((-4, 8)\) by drawing a line and highlighting the region between -4 and 8 with open circles at both end-points. An example sketch is shown below: ``` ------(----------)--------------- -5 -4 8 10 ``` In this sketch, the parentheses at -4 and 8 indicate open circles, meaning the end-points are not included in the solution. The highlighted region between -4 and 8 shows all values of x that satisfy the inequality.

Key Concepts

Solving Inequalities

Inequalities are similar to equations, but instead of an equal sign, they use symbols such as ">", "<", "≥", or "≤". In this problem, the inequality involves an absolute value expression, which requires a unique approach.
Consider \(|6 - 3x| < 18\). Since absolute value denotes distance from zero, the inequality translates to ensuring the expression inside falls within a certain range relative to zero.
For this, we break down the inequality into two cases:

• One where \(6 - 3x\) is positive or zero, leading to \(6 - 3x < 18\).
• The other where \(6 - 3x\) is negative, translating to \(- (6 - 3x) < 18\).

Solving involves handling each situation separately, utilizing algebra to isolate \(x\). Each piece simplifies through basic operations like addition, subtraction, and division. Be mindful if you divide or multiply by a negative number, as you must flip the inequality sign. This step-by-step approach ensures you capture all possibilities of \(x\) satisfying the inequality.
The first case gives \(x > -4\), and the second yields \(x < 8\). Combining these informs us that \(x\) exists between these two values.

Real Number Line

Visualizing inequalities is often aided by sketches or diagrams. The real number line is a key tool here, stretching infinitely in both directions, representing all real numbers as points.
When sketching \((-4, 8)\) on the real number line, our focus is on illustrating which values of \(x\) meet the inequality's conditions:

• Numbers between \(-4\) and \(8\) are marked.
• Open circles at \(-4\) and \(8\) indicate these endpoints are not part of the solution.

Such diagrams provide a clear representation of where solutions lie. The open interval \((-4, 8)\) can also be depicted as a shaded line segment between these points. Sketching inequalities helps check work visually and comprehend the solution set in a straightforward way, reinforcing your understanding of where solutions occur.

Interval Notation

Interval notation is a concise and standardized way to express ranges of numbers. Rather than writing an inequality with cumbersome symbols, interval notation provides a cleaner statement of which values are included.
For the solution \((-4, 8)\), this type of notation uses parentheses to denote a range excluding the endpoints, known as an 'open interval.'

• Use \( (, )\) for solutions excluding endpoints, a hallmark of open intervals.
• A closed interval uses brackets \[ [, ]\] to indicate inclusivity.
• Mixed sets utilize both when solutions include one endpoint but not the other.

Interval notation's simplicity allows mathematicians and pupils alike to easily communicate quantities at a glance. It helps avoid misinterpretations, offering immediate clarity on where solutions lie within an interval, whether they're bounded openly or closed.