Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|2 x-5|=11$$

Step-by-Step Solution

Answer

The solutions to the equation \(|2x-5|=11\) are \(x=8\) and \(x=-3\).

Step 1: Set Up The Two Conditionals

The operation inside the absolute value is \[2x-5\] and it is equal to 11. But since it is an absolute value, the result sqrt{((2x-5)^2)} can also be negative. Therefore, you'll set up two equations to represent these possibilities: \(2x-5=11\) and \(2x-5=-11.\)

Step 2: Solve The First Conditional Equation

Then, you'll solve the first equation \(2x-5=11\) as follows: Add 5 to both sides to isolate \(2x\), so \(2x = 11+5 = 16\). Divide both sides by 2 gives \(x = 16/2 = 8\). So, one solution is \(x=8\).

Step 3: Solve The Second Conditional Equation

Now, solve the second equation \(2x-5=-11\) using similar steps: Add 5 to both sides to get \(2x=-11+5=-6\). Divide both sides by 2 gives \(x=-6/2=-3\). The second solution to the problem is \(x=-3\).

Step 4: Check The Solutions

To ensure the solutions are correct, substitute them back into the original equation: When \(x=8\), \[|2x-5| = |2(8)-5| = |16-5| = |11|\] which holds true, and when \(x=-3\), \[|2x-5| = |2(-3)-5| = |-6-5| = |-11|\] also holds, meaning \(x=-3\) is also a valid solution.

Key Concepts

Solving Equations with Absolute Values

When solving absolute value equations like \(|2x - 5| = 11\), the goal is to find all possible values of \(x\) that make the equation true. Absolute value equations involve expressions where the outcome must be greater than or equal to zero, capturing both positive and negative possibilities. This is because the absolute value refers to the distance a number is from zero on a number line, which is always non-negative.
To properly solve these equations, the given absolute value equation must be transformed into two separate equations. Given \(|a| = b\), one must consider both scenarios: \(a = b\) and \(a = -b\), because both \(b\) and \(-b\) will result in the absolute value \(|b|\).
This approach sets the stage for solving each equation distinctly and ensuring that all potential solutions are considered. This step is crucial in capturing all scenarios, ensuring none are overlooked.

Understanding Conditional Equations

Conditional equations are those that are true only under specific conditions or particular values of the variables involved. When you solve an absolute value equation, you are dealing with a conditional equation where two scenarios emerge based on the nature of absolute values.
When breaking down an equation like \(|2x - 5| = 11\), it is split into two conditional equations: \(2x - 5 = 11\) and \(2x - 5 = -11\). The term "conditional" indicates that each equation is valid only within certain parameters or values that satisfy the original absolute value equation.

• In the first conditional: Adding 5 to both sides, isolate \(2x\):
  \(2x = 16\).
  Dividing by 2 yields \(x = 8\).
• In the second: Similarly, adding 5 to both sides:
  \(2x = -6\).
  Dividing by 2 gives \(x = -3\).

These equations must be solved separately to reveal the full set of solutions to the original problem, highlighting how conditional solutions are rooted in their specific scenarios.

Checking Solutions of Absolute Value Equations

After determining potential solutions to an absolute value equation, it's essential to verify these solutions to ensure they are correct. Checking solutions means substituting these values back into the original equation to see if they hold true.
Using our earlier solutions \(x = 8\) and \(x = -3\), we substitute them into the original equation \(|2x - 5| = 11\):

• For \(x = 8\): Substitute into the equation:
  \(|2 \times 8 - 5| = |16 - 5| = |11|\). This simplifies to \(11\), which matches the right side of the equation, confirming it's correct.
• For \(x = -3\): Substitute similarly:
  \(|2 \times -3 - 5| = |-6 - 5| = |-11|\). Absolute value of \(-11\) is \(11\), again matching the equation's requirement.

By checking each solution this way, you confirm that both \(x = 8\) and \(x = -3\) satisfy the original equation \(|2x - 5| = 11\), proving their validity. This step prevents errors and ensures that the solutions truly solve the equation as intended.