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: Solve Inequality (a)

To solve \(|2x + 3| < 6\), remove the absolute value by setting up two inequalities: 1. Solve \(2x + 3 < 6\), which gives \(2x < 3\), so \(x < \frac{3}{2}\).2. Solve \(2x + 3 > -6\), which gives \(2x > -9\), so \(x > -\frac{9}{2}\).Combine these solutions as \(-\frac{9}{2} < x < \frac{3}{2}\).

Step 2: Solve Inequality (b)

For \(|3 - 4x| \geq 2\), remove the absolute value by setting up two inequalities:1. Solve \(3 - 4x \geq 2\), which gives \(-4x \geq -1\), leading to \(x \leq \frac{1}{4}\).2. Solve \(3 - 4x \leq -2\), which gives \(-4x \leq -5\), leading to \(x \geq \frac{5}{4}\).Combine these as \(x \leq \frac{1}{4}\) or \(x \geq \frac{5}{4}\).

Step 3: Solve Inequality (c)

For \(|x + 5| \leq 1\), set up the corresponding inequalities:1. Solve \(x + 5 \leq 1\), which gives \(x \leq -4\).2. Solve \(x + 5 \geq -1\), which gives \(x \geq -6\).Combine these solutions as \(-6 \leq x \leq -4\).

Step 4: Solve Inequality (d)

The inequality \(|7 - 2x| < 0\) has no solutions. The absolute value \(|\cdot|\) is always non-negative, thus cannot be less than 0. Therefore, this inequality is impossible.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve finding the range of values for a variable within an inequality that includes absolute value expressions. Understanding the absolute value is key. The absolute value of a number, represented as \(|x|\), tells us how far away that number is from zero on the number line, regardless of direction. For example, \(|-3| = 3\).
When dealing with absolute value inequalities, we create two separate inequalities to solve the problem. If you have an inequality like \(|a| < b\), where \(b\) is greater than zero, it translates to two inequalities: \(-b < a < b\).
For instance, with exercise (a), \(|2x + 3| < 6\) translates to \(-6 < 2x + 3 < 6\). You then solve for \(x\) within this compound inequality by separating it into two parts: \(2x + 3 < 6\) and \(2x + 3 > -6\). In the case where the inequality uses \(\geq\) or \(\leq\), such as in (b) \(\|3-4x\| \geq 2\), it splits into \(3 - 4x \geq 2\) or \(3 - 4x \leq -2\). Understanding this setup is crucial for solving such inequalities successfully.

Inequality Solutions

Finding the solution to inequalities involves not just solving for a variable but also understanding the range of values that meet the inequality's conditions. With each inequality, after separating it into two parts based on the absolute value, you solve each regular inequality.
For example, with exercise (a), you found \(2x + 3 < 6\), which simplifies to \(x < \frac{3}{2}\), and \(2x + 3 > -6\), leading to \(x > -\frac{9}{2}\). Combine these results to find the range of \(x\):

• \(-\frac{9}{2} < x < \frac{3}{2}\)

When combining inequalities, if you're dealing with \(or\), as in (b), the solution becomes a union where either part of the inequality holds true. Thus, for \(\|3 – 4x\| \geq 2\), you find intervals of \(x\) for which the inequality is fulfilled:

• \(x \leq \frac{1}{4}\)
• \(x \geq \frac{5}{4}\)

These techniques help you not just find solutions but see the pattern that the absolute value inequality creates on the number line.

Mathematical Problem Solving

Effective mathematical problem solving involves a structured approach. Start by understanding the symbols and expressions in your problem, such as the absolute value, which simplifies inequalities or equations. Breaking down problems step-by-step is essential.

• First, interpret the mathematical symbols correctly. For example, recognizing that an inequality like \(|x+5| \leq 1\) deals with numbers at most 1 unit from -5.
• Next, deconstruct the inequality to solve it as shown in exercises (a), (b), and (c), by forming equations with the boundary values directly from the inequality \(=\) and simplifying them to find possible values of \(x\).
• Learn to identify when an inequality has no solution, as seen in (d) \(|7 - 2x| < 0\); since absolute values are inherently non-negative, such inequalities are impossible.

Each part of the solution process builds upon the previous step, always rechecking your work to ensure solutions are logical and complete.