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Chapter 1: Problem 11

Solve each equation. $$ |2 x-1|=11 $$

Short Answer

Expert verified
The solutions are \( x = 6 \) and \( x = -5 \).

Step by step solution

01

- Understand the Absolute Value

The absolute value equation \( |2x - 1| = 11 \) implies that the expression inside the absolute value can be either positive 11 or negative 11. This gives us two separate equations to solve: \(2x - 1 = 11\) and \(2x - 1 = -11\).
02

- Solve the First Equation

Solve \(2x - 1 = 11\). First, add 1 to both sides to get \(2x = 12\). Then divide both sides by 2 to find \(x = 6\).
03

- Solve the Second Equation

Solve \(2x - 1 = -11\). First, add 1 to both sides to get \(2x = -10\). Then divide both sides by 2 to find \(x = -5\).
04

- Summarize the Solutions

The solutions to the equation \( |2x - 1| = 11 \) are \( x = 6 \) and \( x = -5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Absolute Value
Absolute value represents the distance of a number from zero on a number line, regardless of direction. So, \( \left| x \right| \) is always positive or zero. When solving equations that include absolute values, you need to consider that the expression inside the absolute value can be positive or negative. For example, solving \( \left| 2x - 1 \right| = 11 \) involves finding both \( 2x - 1 = 11 \) and \( 2x - 1 = -11 \). This is because both 11 and -11 are 11 units away from zero.
Solving Linear Equations
Linear equations are equations of the first degree, which means the highest power of the variable is one. Solving a linear equation typically involves isolating the variable on one side of the equation. For instance, to solve \( 2x - 1 = 11 \), you add 1 to both sides (resulting in \( 2x = 12 \)). Then, you divide both sides by 2 to isolate \( x \), giving \( x = 6 \). Similarly, to solve \( 2x - 1 = -11 \), add 1 to both sides to get \( 2x = -10\), and then divide by 2 to find \( x = -5 \). Applying these steps systematically helps in breaking down the equation into manageable pieces.
Understanding Algebraic Equations
Algebraic equations involve variables and constants combined using mathematical operations. In the case of absolute value equations like \( \left| 2x - 1 \right| = 11 \), you transform it into two separate linear equations. Algebraic methods allow us to solve for unknowns by performing operations that maintain equality. These methods include addition, subtraction, multiplication, and division. Properly understanding how these operations work in unison provides a sturdy foundation for solving more complex equations encountered in algebra.

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