Problem 3 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10, solve for \(x\). $$ |5-2 x|=11 $$

Step-by-Step Solution

Answer

The solutions are \( x = -3 \) and \( x = 8 \).

Step 1: Understand the Absolute Value Equation

Recall that the absolute value \( |a| \) represents the distance of \(a\) from zero on a number line. This means \( |5-2x| = 11 \) implies two possible equations: \( 5-2x = 11 \) and \( 5-2x = -11 \).

Step 2: Solve the First Equation

Set up the first equation: \( 5 - 2x = 11 \.\) To solve for \x\, first subtract \(5\) from both sides: \( -2x = 6 \.\) Then, divide by \ -2 \ on both sides: \( x = -3 \).

Step 3: Solve the Second Equation

Set up the second equation: \( 5 - 2x = -11 \.\) To solve for \x\, first subtract \(5\) from both sides: \( -2x = -16 \.\) Then, divide by \ -2 \ on both sides: \( x = 8 \).

Step 4: Write the Solution

Combine the solutions from both equations. The solutions to \( |5-2x| = 11 \) are \( x = -3 \) and \( x = 8 \).

Key Concepts

solving absolute value equations

Absolute value equations are equations where the absolute value of a variable expression is set equal to a number. To solve these equations, it's crucial to understand what absolute value represents.
The absolute value \(|a|\) of a number \(a\) is its distance from zero on the number line, disregarding its sign. Therefore, an equation of the form \(|A| = B\) can be split into two separate equations: \(A = B\) and \(A = -B\).
For example, if you have \(|5-2x|=11\), you must consider both \(5-2x=11\) and \(5-2x=-11\). This step is crucial as it ensures all possible values of \(x\) are found.

equation solving steps

Breaking down the steps of solving an absolute value equation can make the process more straightforward. Let's review the solution to the previous problem step by step:

• Step 1: Understand that \(|5-2x|=11\) translates to setting up two possible equations: \(5-2x=11\) and \(5-2x=-11\).
• Step 2: Solve the first equation \(5-2x=11\). Subtract 5 from both sides to isolate the term with the variable: \(-2x=6\). Then, divide by -2 to solve for \(x\): \(x=-3\).
• Step 3: Solve the second equation \(5-2x=-11\). Similarly, subtract 5 from both sides: \(-2x=-16\). Next, divide by -2: \(x=8\).
• Step 4: Combine the solutions. The solutions to the equation \(|5-2x|=11\) are \(x=-3\) and \(x=8\).

By following these steps systematically, you can solve absolute value equations without confusion.

algebraic solutions

When working with absolute value equations, using algebraic methods is essential.
Initially, separate the absolute value equation into two distinct linear equations. Then, solve each linear equation using basic algebra techniques such as addition, subtraction, multiplication, and division.
For example, take the equation \(|5-2x|=11\). The first linear equation, \(5-2x=11\), simplifies to \(x=-3\) after isolating \(x\). For the second linear equation, \(5-2x=-11\), solving also involves isolating \(x\), resulting in \(x=8\).
Ultimately, ensuring you understand each algebraic step allows you to successfully solve absolute value equations. Over time, practice will make recognizing how to manipulate these equations second nature.