Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Use the algebraic interpretation of absolute value to solve each of the following. Please display your answers on a number line. (a) \(|2 x+1 / 2| \leq 6\) (b) \(|3 x-4|>8\)

Step-by-Step Solution

Answer

Part (a) solution: \(x\) is in the closed interval [\(-13/2, 11/2\)]. Part (b) solution: \(x\) is in the union of the open intervals \((-\infty, -4/3) ∪ (4, ∞)\).

Step 1: Solve Part (a)

To start with, the absolute value of \(2x + 1/2\) must be 'less than or equal to' 6. We have to solve for \(x\) in two cases: case 1) \(2x + 1/2 ≤ 6\) and case 2) \(2x + 1/2 ≥ -6\). Then express your solutions in interval notation and plot them on a number line.

Step 2: Solve Part (b)

Here, the absolute value of \(3x - 4\) must be 'greater than' 8. Similarly, we solve for \(x\) in two cases, this time however case 1) \(3x - 4 > 8\) and case 2) \(3x - 4 < -8\). Once again, we express the solutions in interval notation and plot them on a number line.

Step 3: Perform Calculations for Part (a)

Solve the inequalities in step 1: For case 1): Subtract 1/2 from each side, then divide each side by 2, we get \(x ≤ 11/2\). For case 2): Subtract 1/2 from each side, and then divide each side by 2, we get \(x ≥ -13/2\). These solutions can be plotted as [\(-13/2, 11/2\)] on a number line.

Step 4: Perform Calculations for Part (b)

Solve the inequalities in step 2: For case 1): Add 4 to both sides, then divide each side by 3, we get \(x > 4\). For case 2): Add 4 to both sides, and then divide each side by 3, we get \(x < -4/3\). These solutions can be plotted as \((-\infty, -4/3) ∪ (4, ∞)\) on a number line.

Key Concepts

Algebraic Interpretation

Absolute value inequalities can be solved by understanding the algebraic interpretation of the absolute value function. The absolute value \( |a| \) represents the distance of \( a \) from zero on the number line, without regard to direction.
When you solve an inequality involving absolute value, such as \(|x| \leq 6\), you are finding the range of values for \(x\) that are within a certain distance from zero. Breaking it down into two scenarios helps us understand these boundaries:

• Scenario 1: The expression inside the absolute value is less than or equal to the number, in this case, \(2x + 1/2 \leq 6\).


• Scenario 2: The expression is greater than or equal to the negative of the number, i.e., \(2x + 1/2 \geq -6\).

By solving these two related inequalities, we can determine the full set of values for \( x \) that satisfy the original absolute value inequality.

Interval Notation

When we solve an inequality, the solution is often a range of values, not just a single number. Interval notation provides a compact way to write these ranges.
For instance, if the solution to \(|2x + 1/2| \leq 6\) is \(-13/2 \leq x \leq 11/2\), interval notation expresses this as \([-13/2, 11/2]\).

• Brackets \([ \) and \( ] \) indicate that the endpoint is included in the interval, known as a closed interval.


• Parentheses \(( \) and \( ) \) mean the endpoint is not included, an open interval.

For \(|3x - 4| > 8\), the solution is \((-\infty, -4/3) \cup (4, \infty)\), meaning \(x\) is less than \(-4/3\) or greater than 4. This is useful because it communicates the solution set's properties in a clear and concise manner.

Number Line

A number line is a valuable visual tool to understand and display solutions to absolute value inequalities. It allows us to see at a glance which portions of the number line are included in the solution.
For example, the solution \([-13/2, 11/2]\) for \(|2x + 1/2| \leq 6\) will be represented with a solid line covering this interval, making it clear that all these \(x\) values satisfy the inequality.
For the inequality \(|3x - 4| > 8\), the solution \((-\infty, -4/3) \cup (4, \infty)\) is shown on a number line with two separate sections, one to the left of \(-4/3\) and the other to the right of 4.

• Closed dots or solid lines indicate that an endpoint is included (like brackets \([, ]\) in interval notation).


• Open dots or dashed lines highlight that an endpoint is not included (like parentheses \((, )\) in interval notation).

This visualization helps ensure a more intuitive understanding of where solutions lie on the number line.

Inequality Solving

Solving inequalities involves similar steps to solving equations but with additional rules for maintaining inequality. For absolute value inequalities \(|ax+b| \leq c\) or \(|ax+b| > c\), we define two separate cases.
For \(|2x + 1/2| \leq 6\), set up:

• \(2x + 1/2 \leq 6\)
• \(2x + 1/2 \geq -6\)

Solve each linear inequality to get \(x\)'s range. Take care to maintain the direction of inequalities when multiplying or dividing by negative numbers, as it reverses the inequality sign.
For \(|3x - 4| > 8\), the setups are:

• \(3x - 4 > 8\)
• \(3x - 4 < -8\)

These inequalities are then resolved to find \(x\)'s solution set. It is crucial to approach each step cautiously to preserve the integrity of the inequality, leading to an accurate solution.