Problem 66 - Algebra and Trigonometry

Question

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$|2 x-3|=11$$

Step-by-Step Solution

Answer

The solutions are \(x = 7\) and \(x = -4\)

Step 1: Set Up Two Equations

Given the absolute value equation \(|2x - 3| = 11\), set up two linear equations based on the property of absolute value. The two equations will be: \(2x - 3 = 11\) and \(2x - 3 = -11\).

Step 2: Solve Each Equation

Solve each equation from previous step separately. \n\nFirst equation is \(2x - 3 = 11. Add 3 to both sides of the equation to get: 2x = 14.\ Divide both sides by 2: x = 7.\n\nSecond equation is \(2x - 3 = -11. Add 3 to both sides of the equation to get: 2x = -8. Divide both sides by 2: x = -4.\)

Step 3: Conclusion

The solutions of the absolute value equation |2x - 3| = 11 are the solutions of the two linear equations, which are \(x = 7\) and \(x = -4\).

Key Concepts

Absolute Value Properties

Understanding the absolute value is fundamental in solving equations like \(|2x - 3| = 11\). The absolute value of a number refers to its distance from zero on the number line, without considering which direction from zero the number lies. Therefore, the absolute value is always non-negative. For any real number \(a\), the absolute value is defined as \(|a|\), which equals \(a\) if \(a\) is positive or zero, and \(-a\) if \(a\) is negative.

When dealing with absolute value equations, we must consider both the positive and negative scenarios of the expression inside the absolute value. That's why we set up two separate equations during the solution process—one that considers the inside as a positive quantity and one as a negative quantity. This property allows us to effectively deal with both cases where the absolute value expression could take.

Linear Equations

Linear equations are fundamental to algebra and represent the relationship between two variables in a straight line when plotted on a graph. They take the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants. The solution to a linear equation is the value of \(x\) that makes the equation true. In the context of absolute value equations like the one given, we treat each scenario — the expression being positive or negative — as a separate linear equation.

For example, the given equation \(|2x - 3| = 11\) resulted in two linear equations: \(2x - 3 = 11\) and \(2x - 3 = -11\). Solving a linear equation generally involves isolating the variable, \(x\), by performing basic arithmetic operations such as addition, subtraction, multiplication, and division.

Algebraic Solutions

Algebraic solutions involve finding the values of the variables that satisfy the equation. To solve an algebraic equation, you manipulate the equation using algebraic operations to isolate the variable and solve for its value. The step-by-step procedure in solving the equation \(|2x - 3| = 11\) involved first setting up two equations without the absolute value to represent the two possible values of the expression inside the absolute value.

Consistency in Steps

It is important to be consistent with each step to avoid confusion. That way, we separately added 3 to both sides of the two equations and then divided by 2 to solve for \(x\). This process is central to algebra and is used across various types of equations beyond just absolute value equations.