Problem 66 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: Split the Equation

We split the equation into two separate equations. The first equation is \(2x - 3 = 11\) and the second equation \(2x - 3 = -11\).

Step 2: Solve the First Equation

To solve \(2x - 3 = 11\), first add 3 to both sides of the equation to get \(2x = 14\). Then, divide both sides by 2 to solve for x, yielding \(x = 7\).

Step 3: Solve the Second Equation

To solve \(2x - 3 = -11\), again add 3 to both sides of the equation to get \(2x = -8\). Then divide both sides by 2 to solve for x, giving \(x = -4\).

Key Concepts

Solving Absolute Value Equations

Understanding how to solve absolute value equations is a crucial step in algebra. Absolute value represents the distance of a number from zero on the number line, regardless of direction. Because of this, absolute value equations often split into two different equations.

In our example, the equation \(|2x - 3| = 11\) involves transforming the absolute value into two separate scenarios: one where the expression inside the absolute value is positive (\(2x - 3 = 11\)) and another where it is negative (\(2x - 3 = -11\)). This is because both expressions, \(11\) and \(-11\), when absolute, give the distance of \(11\) from zero.

This split approach captures all possible solutions, ensuring no potential answer is missed.

The Role of Algebra in Solving Equations

Algebra allows us to manipulate equations to find unknown values. In this problem, algebraic manipulation helps isolate the variable \(x\). We start with the equations

• \(2x - 3 = 11\)
• \(2x - 3 = -11\)

In both instances, adding \(3\) to each side prepares the equation for the next step, which is to isolate \(x\) by dividing all terms by \(2\).

These steps illustrate how algebra can be applied to rearrange equations, maintaining equality while solving for unknowns. Every operation done to one side of the equation is mimicked on the other side, preserving the balance of the equation.

This fundamental element of algebra ensures that the solutions we find are correct and applicable.

Mathematical Solutions and Their Verification

Once we've found our solutions, verifying them ensures their correctness. From the problem, we derived two potential solutions: \(x = 7\) and \(x = -4\). These need to be checked with the original absolute value equation to confirm they provide true statements:

Substitute \(x = 7\) back into the original equation, resulting in \(|2(7) - 3| = |14 - 3| = |11| = 11\), which satisfies the equation. Similarly, substituting \(x = -4\) results in \(|2(-4) - 3| = |-8 - 3| = |-11| = 11\), also satisfying the equation.

This verification process confirms that both solutions are indeed valid. By substituting back into the original equation, we validate the functionality and reliability of our algebraic manipulations, ensuring no errors were made in the solving process.