Problem 3 - Calculus for Biology and Medicine

Question

Solve the following equations: (a) \(|2 x-4|=6\) (b) \(|x-3|=2\) (c) \(|2 x+3|=5\) (d) \(|7-3 x|=-2\)

Step-by-Step Solution

Answer

(a) x = 5, x = -1; (b) x = 5, x = 1; (c) x = 1, x = -4; (d) No solution.

Step 1: Understanding Absolute Values

The key to solving equations involving absolute values is to remember that the absolute value of a number is its distance from zero on the number line, which is always non-negative. Thus, for an equation of the form \(|A| = B\), where \(B\) is a non-negative number, we solve two separate equations: \(A = B\) and \(A = -B\). If \(B\) is negative, there are no solutions, since absolute values cannot be negative.

Step 2: Solving Part (a)

For the equation \(|2x - 4| = 6\), set up two equations: 1. Solve \(2x - 4 = 6\): - Add 4 to both sides: \(2x = 10\) - Divide by 2: \(x = 5\)2. Solve \(2x - 4 = -6\): - Add 4 to both sides: \(2x = -2\) - Divide by 2: \(x = -1\)The solutions are \(x = 5\) and \(x = -1\).

Step 3: Solving Part (b)

For the equation \(|x - 3| = 2\), we set up two equations: 1. Solve \(x - 3 = 2\): - Add 3 to both sides: \(x = 5\)2. Solve \(x - 3 = -2\): - Add 3 to both sides: \(x = 1\)The solutions are \(x = 5\) and \(x = 1\).

Step 4: Solving Part (c)

For the equation \(|2x + 3| = 5\), set up two equations: 1. Solve \(2x + 3 = 5\): - Subtract 3 from both sides: \(2x = 2\) - Divide by 2: \(x = 1\)2. Solve \(2x + 3 = -5\): - Subtract 3 from both sides: \(2x = -8\) - Divide by 2: \(x = -4\)The solutions are \(x = 1\) and \(x = -4\).

Step 5: Analyzing Part (d)

For the equation \(|7 - 3x| = -2\), notice that the equation is not solvable because the right side of the equation is negative, while an absolute value can only be zero or positive.

Key Concepts

Solving Equations

Solving equations, especially those involving absolute values, can initially seem tricky, but by breaking it into clear steps, it's entirely manageable. When faced with an equation like \(|A| = B\), where \(|A|\) represents the absolute value, the concept is to consider the distance of \(A\) from zero on a number line. If \(B\) is a positive number, we need to account for two possibilities because absolute value simply denotes magnitude without considering direction:

• The expression inside the absolute value equals the positive value: \(A = B\).
• The expression inside the absolute value equals the negative of the positive value: \(A = -B\).

For example, with an equation like \(|2x - 4| = 6\), this method leads us to solve separately for \(2x - 4 = 6\) and \(2x - 4 = -6\), giving potential solutions. The deeper understanding of absolute value properties is essential here:
They are always non-negative, implying any problem setup must reflect this, as seen in instances where no solutions are possible if the absolute value is set equal to a negative number.

Mathematics

Mathematics is the study of numbers, quantities, and shapes, which finds its basis in structure, order, and relation. In the realm of mathematics, solving absolute value equations highlights the essence of mathematical reasoning and logic. Through abstraction and logical deduction, it teaches us to approach problems methodically.
In our exercise, by identifying equations like \( |x - 3| = 2 \), we are dealing with a search for values that satisfy a defined condition - 'closeness' to zero on the number line. Mathematics encourages a systematic dissection of these problems, reinforcing logical thinking and problem-solving skills.

• It involves simplifying expressions,
• extracting meaningful conclusions from mathematical truths,
• and verifying through substitution the validity of our attempted solutions.

Thus, diving into such problems is more than just finding a solution; it's an exploration of mathematical principles that govern everyday logic and decision-making.

Algebra

Algebra serves as a critical language in mathematics. It deals with symbols and the rules for manipulating these symbols, allowing us to create formulaic and general mathematical statements. Solving algebraic equations, like absolute value equations, ties directly into early algebraic principles.
In problems like \(|2x + 3| = 5\), algebra requires us to open up two possible scenarios, positioning our equation both in a true positive and a reflected negative light. This simultaneous consideration forms the crux of solving absolute value equations. Through algebra, we learn to:

• Manipulate equations by performing operations equally on both sides to maintain balance.
• Address more complex representations of \(x\) by expanding or simplifying terms.
• Discover solutions that aptly satisfy the equation based on algebraic principles.

Understanding algebra paves the way for further exploration of mathematical concepts, as it helps develop the ability to think abstractly and solve real-world problems symbolically.