Problem 5 - Calculus for Biology and Medicine

Question

Solve the following inequalities: (a) \(|5 x-2| \leq 4\) (b) \(|1-3 x|>8\) (c) \(|7 x+4| \geq 3\) (d) \(|6-5 x|<7\)

Step-by-Step Solution

Answer

(a) \(-\frac{2}{5} \leq x \leq \frac{6}{5}\), (b) \(x < -\frac{7}{3}\) or \(x > 3\), (c) \(x \geq -\frac{1}{7}\) or \(x \leq -1\), (d) \(-\frac{1}{5} < x < \frac{13}{5}\).

Step 1: Understanding the Absolute Value Inequality

To solve an inequality involving an absolute value, such as \(|ax+b| \leq c\), understand that this splits into two cases: \(-c \leq ax+b \leq c\). For \(|ax+b|>c\), the cases are \(ax+b>c\) or \(ax+b<-c\). Similarly, \(|ax+b| \geq c\) results in \(ax+b \geq c\) or \(ax+b \leq -c\).

Step 2: Case (a): Solve \(|5x-2| \leq 4\)

First, write the inequality as two separate inequalities: \(-4 \leq 5x-2 \leq 4\). Solve for \(x\): 1. First inequality: \(-4 \leq 5x-2\) Add 2 to both sides: \(-2 \leq 5x\) Divide by 5: \(-\frac{2}{5} \leq x\).2. Second inequality: \(5x-2 \leq 4\) Add 2 to both sides: \(5x \leq 6\) Divide by 5: \(x \leq \frac{6}{5}\).Therefore, the solution is \(-\frac{2}{5} \leq x \leq \frac{6}{5}\).

Step 3: Case (b): Solve \(|1-3x|>8\)

This splits into two inequalities: 1. \(1-3x>8\) Subtract 1 from both sides: \(-3x>7\) Divide by \(-3\) (reverse the inequality): \(x<-\frac{7}{3}\).2. \(1-3x<-8\) Subtract 1 from both sides: \(-3x<-9\) Divide by \(-3\) (reverse the inequality): \(x>3\).So the solution is \(x<-\frac{7}{3}\) or \(x>3\).

Step 4: Case (c): Solve \(|7x+4| \geq 3\)

This results in two inequalities:1. \(7x+4 \geq 3\) Subtract 4 from both sides: \(7x \geq -1\) Divide by 7: \(x \geq -\frac{1}{7}\).2. \(7x+4 \leq -3\) Subtract 4 from both sides: \(7x \leq -7\) Divide by 7: \(x \leq -1\).The solution is \(x \geq -\frac{1}{7}\) or \(x \leq -1\).

Step 5: Case (d): Solve \(|6-5x|<7\)

Split it into two inequalities: \(-7 < 6-5x < 7\). Solve as follows:1. First inequality: \(-7 < 6-5x\) Subtract 6 from both sides: \(-13 < -5x\) Divide by \(-5\) (reverse the inequality): \(x < \frac{13}{5}\).2. Second inequality: \(6-5x < 7\) Subtract 6 from both sides: \(-5x < 1\) Divide by \(-5\) (reverse the inequality): \(x > -\frac{1}{5}\).Therefore, the solution for this case is \(-\frac{1}{5} < x < \frac{13}{5}\).

Key Concepts

Solving Inequalities

When solving inequalities, the objective is to find a range of values that satisfy the given inequality. Unlike equations, where you're looking for specific values, here you're finding a set or interval of numbers. Let's take an example: if you have the inequality \( x + 3 > 5 \), you're interested in determining which values of \( x \) make this statement true.
To solve it:

• Subtract 3 from both sides to isolate \( x \).
• This gives you \( x > 2 \), so any number greater than 2 will work.

What makes inequalities unique is that solutions can often be non-exclusive or inclusive ranges. This difference is crucial for understanding more complex cases, especially when dealing with absolute values.

Case Analysis

Case analysis is a powerful approach for tackling inequalities involving absolute values. Absolute value expresses the distance of a number from zero, so \(|x| = a\) means \(x\) is \(a\) units away from zero, either positively or negatively. Hence, it involves two scenarios.
Consider a typical absolute value inequality like \(|x - 5| \leq 3\). This implies two cases that you need to examine simultaneously:

• Case 1: \( x - 5 \leq 3 \)
• Case 2: \( x - 5 \geq -3 \)

By solving these inequalities separately, you can find a common solution that fits both cases. The result will be a range of values that satisfy the absolute value inequality, providing a full picture of the solution.

Reverse Inequality

Reversing inequalities occurs when you multiply or divide both sides of an inequality by a negative number. This is a critical rule to remember because it works differently than when dealing with equations.
For example, if you have \(-3x > 6\), and you decide to divide both sides by \(-3\) to solve for \(x\), the inequality sign flips. So, \(x < -2\).
Here's a step-by-step process for reversing inequalities:

• Identify when multiplying or dividing by a negative number is necessary.
• After the operation, flip the inequality sign to maintain a true statement.
• Continue solving like normal equations from there.

This rule is often encountered in solving absolute value inequalities, as seen in the process of removing absolute value bars and solving the resulting expressions.

Splitting Absolute Value Inequalities

Splitting absolute value inequalities involves rewriting them into a form that can be solved using regular inequality techniques. Absolute value inequalities, such as \( |ax+b| < c \), imply a range without explicitly showing it at first glance.
These are split into logical parts:

• If \( |ax+b| \leq c \), consider \(-c \leq ax+b \leq c\).
• If \( |ax+b| > c \), use separate conditions: \( ax+b > c \quad\text{or}\quad ax+b < -c \).

This method allows you to handle complex expressions by reducing them into simpler, manageable parts. By addressing each part individually, you ensure the correct solution set for the inequality, making the process structured and straightforward.