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) \(|1-5 x|=6\)

Step-by-Step Solution

Answer

(a) x = 1, x = -5; (b) x = 5, x = 1; (c) x = 4, x = -1; (d) x = -1, x = 7/5.

Step 1: Understand Absolute Value Equations

An absolute value equation, such as \(|a|=b\), implies two possible equations: \(a = b\) or \(a = -b\). This is because the absolute value of a number represents its distance from zero on the number line, and distance is always positive.

Step 2: Solve Equation (a) |2x+4|=6

For this equation, split it into two cases:1. \(2x + 4 = 6\)2. \(2x + 4 = -6\)For the first case, solve: \(2x + 4 = 6 \rightarrow 2x = 2 \rightarrow x = 1\)For the second case, solve:\(2x + 4 = -6 \rightarrow 2x = -10 \rightarrow x = -5\)The solutions are \(x = 1\) and \(x = -5\).

Step 3: Solve Equation (b) |x-3|=2

For this equation, split it into two cases:1. \(x - 3 = 2\)2. \(x - 3 = -2\)For the first case, solve:\(x - 3 = 2 \rightarrow x = 5\)For the second case, solve:\(x - 3 = -2 \rightarrow x = 1\)The solutions are \(x = 5\) and \(x = 1\).

Step 4: Solve Equation (c) |2x-3|=5

For this equation, split it into two cases:1. \(2x - 3 = 5\)2. \(2x - 3 = -5\)For the first case, solve:\(2x - 3 = 5 \rightarrow 2x = 8 \rightarrow x = 4\)For the second case, solve:\(2x - 3 = -5 \rightarrow 2x = -2 \rightarrow x = -1\)The solutions are \(x = 4\) and \(x = -1\).

Step 5: Solve Equation (d) |1-5x|=6

For this equation, split it into two cases:1. \(1 - 5x = 6\)2. \(1 - 5x = -6\)For the first case, solve:\(1 - 5x = 6 \rightarrow -5x = 5 \rightarrow x = -1\)For the second case, solve:\(1 - 5x = -6 \rightarrow -5x = -7 \rightarrow x = \frac{7}{5}\)The solutions are \(x = -1\) and \(x = \frac{7}{5}\).

Key Concepts

Mathematical Solutions

Mathematical solutions often involve manipulating equations to find unknown variables. An absolute value equation transforms into two separate linear equations. The reason is that the expression inside the absolute value can represent two situations - either positive or negative, relative to the number itself. For instance, with \( |a| = b \), you get \( a = b \) and \( a = -b \). This dual possibility respects the nature of absolute values, which capture a number's distance from zero. Hence, solutions to these equations come from solving each of these possibilities separately.

Step-by-step Problem Solving

Step-by-step problem solving is a reliable approach to tackle equations on paper. When dealing with absolute value equations, like \( |2x+4|=6 \), the process is quite methodical:

• Identify the absolute value equation and note the expression within the absolute value and the constant it is equal to.
• Write down two separate linear equations, removing the absolute value: one equal to the positive version of the constant, the other equal to its negative.
• Solve each of these linear equations individually, following algebraic steps like isolation of the variable.
• Verify your solutions by plugging them back into the original equation to ensure they satisfy the absolute value condition.

This step-by-step approach reduces complex equations to manageable steps and ensures that no solution path is overlooked.

Distance on Number Line

The concept of distance on a number line is essential to understanding absolute values. Absolute value measures how far a number is located from zero, irrespective of the direction. Think of \( |x| \) as asking, "How many units is \( x \) away from zero?"

• If \( x \) is positive, the distance is simply \( x \).
• If \( x \) is negative, the distance is still \( x \) in magnitude but opposite in sign, thus the absolute value \( |x| \) becomes positive.
• This implies a number's absolute value is always non-negative.

This visualization helps students see why solving \( |2x-3|=5 \) yields both a positive and negative equation: \( 2x-3=5 \) and \( 2x-3=-5 \). Using distances, it becomes obvious why each solution is valid given the broader number line context.