Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, solve the equation involving absolute value. $$ |2 x-1|-7=-2 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3\) and \(x = -2\).

Step 1: Isolate Absolute Value Expression

First, we need to isolate the absolute value expression on one side of the equation. In the equation \(|2x-1|-7=-2\), we add 7 to both sides to get the absolute value expression by itself. This gives us:\[ |2x-1| = -2 + 7 \]\[ |2x-1| = 5 \]

Step 2: Set Up Two Equations

The absolute value equation \(|a| = b\) implies two separate equations: \(a = b\) and \(a = -b\). Applying this to our expression, \(|2x-1| = 5\) means we write:\[ 2x - 1 = 5 \]\[ 2x - 1 = -5 \]

Step 3: Solve the First Equation

Solve the first equation, \(2x - 1 = 5\). Add 1 to both sides:\[ 2x = 5 + 1 \]\[ 2x = 6 \]Now divide by 2 to solve for \(x\):\[ x = \frac{6}{2} \]\[ x = 3 \]

Step 4: Solve the Second Equation

Now solve the second equation, \(2x - 1 = -5\). Add 1 to both sides:\[ 2x = -5 + 1 \]\[ 2x = -4 \]Now divide by 2 to solve for \(x\):\[ x = \frac{-4}{2} \]\[ x = -2 \]

Step 5: Validate Solutions

Substitute each solution back into the original equation to ensure they satisfy it. For \(x = 3\) and \(x = -2\):- For \(x = 3\): \[ |2(3)-1|-7 = |6-1|-7 = |5|-7 = 5-7 = -2 \] ✔️- For \(x = -2\): \[ |2(-2)-1|-7 = |-4-1|-7 = |-5|-7 = 5-7 = -2 \] ✔️Both solutions satisfy the original equation.

Key Concepts

Solving Equations

When approaching an equation, especially one with absolute values, it helps to start by understanding the structure and goal. The primary aim is to find the values of the variable that make the equation true. Absolute value equations usually have two possible scenarios, stemming from the nature of absolute values. The absolute value of a number is always non-negative, revealing two cases to consider in an equation.
In problems involving absolute value, like \(|2x-1|-7=-2\), first, you isolate the absolute value component. Our aim is to treat the absolute value like a regular algebraic expression, leading to two separate yet related equations to solve. In this context, each scenario represents a potential solution.

Isolating Absolute Value

To effectively solve an equation with an absolute value, one first needs to isolate the absolute value expression. Consider the equation \(|2x-1|-7=-2\).
To isolate \(|2x-1|\), you add 7 to both sides of the equation, eliminating any impediments to isolating the absolute value. This adjustment transforms the equation to:
\[ |2x-1|=5 \]
Having isolated the absolute value, you can then proceed to examine the standard form, which dictates that the absolute value is equal to a specific number. This required rearrangement brings us to the critical next step, setting up two possible equations for further resolution.

Substituting Solutions

Substitution is a process of verifying potential solutions. First, after isolating and setting up two equations derived from \(|2x-1|=5\), you separate the absolute value into \(2x-1=5\) and \(2x-1=-5\).
Solve these equations separately to find potential values for \(x\). This yields:

• For \(2x-1=5\): \(x=3\)
• For \(2x-1=-5\): \(x=-2\)

Each potential solution needs to be checked by substituting back into the original problem. This ensures both solutions are correct and applicable, confirming their viability. It's essential to ensure that the solutions satisfy the transformed absolute value equation as initially set up.

Validating Solutions

Validation confirms the legitimacy of your solutions, ensuring they satisfy the original equation. Start by substituting each solution back into the original equation \(|2x-1|-7=-2\). The goal is to see if both sides return equal values.
For \(x=3\):

• Calculate \(|2(3)-1|-7 =|5|-7\).
• Result is \(-2\), which equals the initial equation’s right-hand side.

For \(x=-2\):

• Calculate \(|2(-2)-1|-7=|-5|-7\).
• Result is \(-2\), consistent with the equation’s requirement.

Completing these steps ensures you've accurately solved both parts of the absolute value equation. Validation is critical to demonstrate the consistency and correctness of your solutions, thereby verifying their legitimacy.