Problem 6 - Calculus for Biology and Medicine

Question

Solve the following inequalities: (a) \(|2 x+3|<6\) (b) \(|3-4 x| \geq 2\) (c) \(|x+5| \leq 1\) (d) \(|7-2 x|<0\)

Step-by-Step Solution

Answer

(a) \(-\frac{9}{2} < x < \frac{3}{2}\), (b) \(x \leq \frac{1}{4}\) or \(x \geq \frac{5}{4}\), (c) \(-6 \leq x \leq -4\), (d) No solution.

Step 1: Understanding Absolute Inequalities

To solve absolute inequalities, we must consider two scenarios for each inequality: one where the expression inside the absolute value is positive, and another where it is negative. Each scenario leads to a different inequality.

Step 2: Solving Part (a)

For \(|2x + 3| < 6\),consider two cases:1. The expression is positive: \(2x + 3 < 6\)2. The expression is negative: \(-(2x + 3) < 6\).Solve each inequality separately.**Case 1:**\(2x + 3 < 6\) becomes \(2x < 3\), dividing both sides by 2 gives:\(x < \frac{3}{2}\).**Case 2:**\(2x + 3 > -6\), rearranges as \(2x > -9\) and dividing both sides by 2 gives:\(x > -\frac{9}{2}\).Combine the results: \(-\frac{9}{2} < x < \frac{3}{2}\).

Step 3: Solving Part (b)

For \(|3 - 4x| \geq 2\):1. Positive scenario: \(3 - 4x \geq 2\)2. Negative scenario: \(-(3 - 4x) \geq 2\) which simplifies to \(3 - 4x \leq -2\).Solve each inequality.**Case 1:**\(3 - 4x \geq 2\) becomes \(-4x \geq -1\), dividing by -4 (and flipping the inequality) gives:\(x \leq \frac{1}{4}\).**Case 2:**\(3 - 4x \leq -2\) becomes \(-4x \leq -5\), dividing by -4 (and flipping the inequality) gives:\(x \geq \frac{5}{4}\).Combine using OR for inequalities:\(x \leq \frac{1}{4}\) OR \(x \geq \frac{5}{4}\).

Step 4: Solving Part (c)

For \(|x + 5| \leq 1\):1. For positive scenario: \(x + 5 \leq 1\)2. For negative scenario: \(x + 5 \geq -1\).Solve each:**Case 1:**\(x + 5 \leq 1\), which gives\(x \leq -4\).**Case 2:**\(x + 5 \geq -1\), which gives \(x \geq -6\).Combine results:\(-6 \leq x \leq -4\).

Step 5: Solving Part (d)

Examining inequality \(|7 - 2x| < 0\): absolute value expressions are always non-negative. No value of \(x\) will satisfy \(|7 - 2x| < 0\). Therefore, there is no solution.

Key Concepts

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, regardless of direction. It is represented by two vertical bars, like \(|x|\). For example, both \(|5|\) and \(-5|\) equal 5. This is because distance cannot be negative.

When dealing with absolute values in inequalities, you're handling two potential situations due to the nature of absolute values:

• When the expression inside the absolute value is non-negative.
• When the expression inside the absolute value is negative (meaning you take the negative of the expression).

This dichotomy forms the basis of how we solve absolute value inequalities.

Inequality Solving

Solving inequalities is similar to solving equations, with some added rules due to the inequality sign. When tackling absolute inequalities, like those presented, we perform the following steps:

• First, break down the absolute inequality into two separate linear inequalities to handle the positive and negative scenarios of the expression inside the absolute value.
• Next, solve these inequalities independently to find the range of solutions.
• Remember: when you multiply or divide both sides of an inequality by a negative number, the inequality sign flips. This is crucial to getting the right solution.

Through these steps, you can accurately depict the set of values (solutions) that satisfy the original absolute inequality.

Inequality Scenarios

Different scenarios arise from absolute inequalities depending on their form, which means you often get two cases to solve. Let's break this down with the inequalities from our problem:

For example in \( |2x + 3| < 6 \), the inequality is split into two cases based on whether \(2x + 3\) is positive or negative. These cases yield an intersection of results representing a range on the number line.

• Case 1: Consider \( 2x + 3 < 6 \)
• Case 2: Consider \(- (2x + 3) < 6 \)

Conversely, inequalities like \( |3 - 4x| \geq 2 \) can be solved by handling the expression similarly, yet the solution represents an ‘or’ scenario on the number line. This signifies disjunct ranges: either value can satisfy the inequality. Intervals may not be continuous, contrasting the \(|3 - 4x|\geq2\) scenario where solutions span two non-overlapping segments.