Problem 64 - College Algebra

Question

Solve each absolute value equation or indicate the equation has no solution. $$ |2 x-3|=11 $$

Step-by-Step Solution

Answer

The solutions for the equation are \(x=7\) and \(x=-4\).

Step 1: Remove Absolute Value

To remove the absolute value, set up two separate equations, one for the positive case and one for the negative case.\n (1) \(2x - 3 = 11\)\n (2) \(2x - 3 = -11\)

Step 2: Solve the Positive Case

Solve the first equation for x. \n \((2x - 3 = 11) ➔ (2x = 14) ➔ (x = 7)\)

Step 3: Solve the Negative Case

Solve the second equation for x. \n\((2x - 3 = -11) ➔ (2x = -8) ➔ (x = -4)\)

Key Concepts

Understanding Absolute Value Equations

When tackling absolute value equations, understanding the fundamental concept of absolute value is crucial. Absolute value refers to the distance of a number from zero on the number line, regardless of direction. Consequently, the absolute value of a number is always non-negative.

For the equation \( |2x-3| = 11 \), the absolute value symbol encompasses a linear expression. The key is to recognize that this equation actually represents two scenarios: when \( 2x-3 \) is positive and equals 11, and when \( 2x-3 \) is negative and its absolute value equals 11. We remove the absolute value by setting up two separate equations to cover both cases for \( x \).

This dual nature of absolute value equations must be considered to find all valid solutions, ensuring a full understanding of possible outcomes for \( x \) that satisfy the original equation.

Solving Linear Equations

Linear equations are algebraic statements of the first degree, meaning they involve variables raised only to the power of one. These equations form a straight line when graphed and are the building blocks for more complex algebraic concepts.

Solving a linear equation, such as \( 2x - 3 = 11 \) or \( 2x - 3 = -11 \) from the absolute value setup, requires isolating the variable. This is achieved by performing inverse operations to 'undo' what has been done to the variable. Remembering to keep the equation balanced by performing the same operations on both sides is imperative. In our example, by adding 3 to both sides and then dividing by 2, we obtain solutions for \( x \) in both cases. This process illustrates the clarity and directness of solving linear equations.

Algebraic Solutions

Finding algebraic solutions entails working through equations step-by-step to uncover the value of the unknowns. It's paramount in algebra to follow systematic procedures to arrive at the correct solutions.

In our absolute value equation example, algebra empowers us to solve for \( x \) by creating and subsequently solving two distinct linear equations. The first equation, \( 2x - 3 = 11 \), gives us a solution when we perform the steps of adding 3 to both sides and then dividing by 2, resulting in \( x = 7 \). The second equation, \( 2x - 3 = -11 \), gives us the solution \( x = -4 \) after similar steps are applied. These algebraic solutions are both important, allowing us to understand the full scope of answers to the original absolute value equation.