Problem 5 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

(a) \(-\frac{2}{5} \leq x \leq \frac{6}{5}\); (b) \(x < -\frac{5}{4}\) or \(x > \frac{11}{4}\); (c) \(x \leq -1\) or \(x \geq -\frac{1}{7}\); (d) \(-5 < x < 2\).

Step 1: Understanding Absolute Value Inequalities

Absolute value inequalities can be broken down into two separate inequalities. For \(|x| \leq a\), the inequality becomes \(-a \leq x \leq a\); for \(|x| \geq a\), it becomes \(x \leq -a\) or \(x \geq a\). For \(|x| > a\), it is captured by \(x < -a\) or \(x > a\).

Step 2: Solving Inequality (a)

Given \(|5x-2| \leq 4\), this transforms into the compound inequality \(-4 \leq 5x - 2 \leq 4\). Solving it requires breaking it into two separate inequalities: \(-4 \leq 5x - 2\) and \(5x - 2 \leq 4\).

Step 3: Solving Part 1 of (a)

For \(-4 \leq 5x - 2\), add 2 to both sides to get \(-2 \leq 5x\). Divide by 5 to get \(-\frac{2}{5} \leq x\).

Step 4: Solving Part 2 of (a)

For \(5x - 2 \leq 4\), add 2 to both sides to get \(5x \leq 6\). Divide by 5 to get \(x \leq \frac{6}{5}\).

Step 5: Combine Results of (a)

The solution for \(|5x-2| \leq 4\) is \(-\frac{2}{5} \leq x \leq \frac{6}{5}\).

Step 6: Solving Inequality (b)

\(|3 - 4x| > 8\) transforms into two separate inequalities: \(3 - 4x < -8\) or \(3 - 4x > 8\).

Step 7: Solving Part 1 of (b)

For \(3 - 4x < -8\), subtract 3 to get \(-4x < -11\). Divide by \(-4\) (remember to reverse the inequality sign) to get \(x > \frac{11}{4}\).

Step 8: Solving Part 2 of (b)

For \(3 - 4x > 8\), subtract 3 to get \(-4x > 5\). Divide by \(-4\) (remember to reverse the inequality sign) to get \(x < -\frac{5}{4}\).

Step 9: Combine Results of (b)

The solution is \(x < -\frac{5}{4}\) or \(x > \frac{11}{4}\).

Step 10: Solving Inequality (c)

\(|7x + 4| \geq 3\) transforms into \(7x + 4 \leq -3\) or \(7x + 4 \geq 3\).

Step 11: Solving Part 1 of (c)

For \(7x + 4 \leq -3\), subtract 4 to get \(7x \leq -7\). Divide by 7 to get \(x \leq -1\).

Step 12: Solving Part 2 of (c)

For \(7x + 4 \geq 3\), subtract 4 to get \(7x \geq -1\). Divide by 7 to get \(x \geq -\frac{1}{7}\).

Step 13: Combine Results of (c)

The solution is \(x \leq -1\) or \(x \geq -\frac{1}{7}\).

Step 14: Solving Inequality (d)

\(|3 + 2x| < 7\) becomes \(-7 < 3 + 2x < 7\).

Step 15: Solving Part 1 of (d)

For \(-7 < 3 + 2x\), subtract 3 from both sides to get \(-10 < 2x\). Divide by 2 to get \(-5 < x\).

Step 16: Solving Part 2 of (d)

For \(3 + 2x < 7\), subtract 3 from both sides to get \(2x < 4\). Divide by 2 to get \(x < 2\).

Step 17: Combine Results of (d)

The solution is \(-5 < x < 2\).

Key Concepts

Compound Inequalities

Compound inequalities involve two separate inequalities connected by the words "and" or "or." To solve a compound inequality, you need to find a solution set that makes both parts true when connected by "and" or at least one true when connected by "or."

• "And" Compound Inequality: These are often expressed as a range, like \( a < x < b \), indicating that the variable meets both conditions simultaneously. For example, combining the steps for solving \(|5x - 2| \leq 4 \) results in a single statement: \(-\frac{2}{5} \leq x \leq \frac{6}{5}\).
• "Or" Compound Inequality: Use terms like \( x < a \) or \( x > b \). For \(|3 - 4x| > 8 \), resulting solutions are separate, i.e., \( x < -\frac{5}{4} \) or \( x > \frac{11}{4} \), meaning any value for \(x\) from these ranges satisfies the inequality.

Understanding compound inequalities is crucial, as they provide a way to express many solutions on a number line.

Solving Inequalities

Solving inequalities is similar to solving equations, but there is a key difference: you must remember to reverse the inequality sign if you multiply or divide by a negative number. Let's go through the general steps involved:
1. Isolate the Variable: Use inverse operations such as addition, subtraction, multiplication, or division to isolate the variable on one side of the inequality.

• For \(|7x + 4| \geq 3 \), you turn this into \(7x + 4 \leq -3\) or \(7x + 4 \geq 3 \), altering the equation.
• Performing operations correctly maintains the integrity of the solution. For instance, dividing \(-4x < -11\) by \(-4\), the inequality becomes \(x > \frac{11}{4}\).

2. Solve Each Resulting Inequality: After breaking down an absolute value inequality, solve each related inequality separately and then combine the results based on whether it's an "and" or "or" compound inequality.
Practicing solving inequalities will keep these rules and steps fresh in your mind.

Absolute Value Properties

Understanding absolute value properties is essential when dealing with inequalities, as they dictate the structure of the resulting inequalities:

• Absolute Value Definition: Absolute value represents the distance a number is from zero on a number line. Hence, it is always a non-negative value.
• Handling \( |x| \leq a \): This results in a compound inequality \(-a \leq x \leq a \), capturing all numbers between \(-a\) and \(a\), inclusive.
• Handling \( |x| > a \): This splits into two cases: \(x < -a \) or \(x > a \), identifying numbers beyond either side of \(-a\) and \(a\) on the number line.

Utilizing absolute value properties allows one to shift from a single, more abstract concept into manageable inequalities, leading to easier solutions.