Problem 51 - Intermediate Algebra

Question

Solve each equation. See Example 5. $$ |2-x|=|3 x+2| $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -2\).

Step 1: Set Up Different Cases

Since we are dealing with absolute values, we need to consider the cases where the expressions inside the absolute value function take positive and negative values. We create two equations to account for these possibilities:1. Set both expressions equal: \[2 - x = 3x + 2\]2. Set one expression equal to the negative of the other:\[2 - x = -(3x + 2)\] which simplifies to \[2 - x = -3x - 2\].

Step 2: Solve First Case

Let's solve the first case where both expressions within the absolute values are equal:\[2 - x = 3x + 2\]Rearrange the terms to solve for \(x\):\[2 - 2 = 3x + x\]\[0 = 4x\]\[x = 0\].

Step 3: Solve Second Case

Now solve the second case where one expression is the negative of the other:\[2 - x = -3x - 2\]Rearrange to collect all \(x\) terms on one side and constants on the other:\[2 + 2 = -3x + x\]\[4 = -2x\]\[x = -2\].

Step 4: Check Solutions

Verify the solutions by plugging them back into the original equation:1. Check \(x = 0\): \[|2 - 0| = |3(0) + 2|\] \[2 = 2\] which is true.2. Check \(x = -2\): \[|2 - (-2)| = |3(-2) + 2|\] \[|4| = |-4|\] \[4 = 4\] which is also true.

Key Concepts

Case Analysis in Absolute Value Equations

When dealing with absolute value equations like \(|2-x| = |3x+2|\), it's important to remember that absolute values measure distance from zero. As such, what's inside the absolute value signs can be positive or negative—leading to different scenarios that need consideration. This is where case analysis becomes critical.

• Start by identifying the equations involved: in this instance, \(2-x\) and \(3x+2\).
• Set up two scenarios: One where both sides of the equation are equal, and one where they are negatives of each other.

This process helps capture all potential solutions, as any real number could result from equating these values. Think of it as trying to "catch all the fish" in a mathematical net, ensuring no possibilities slip by.

Solving the Equations

Once you've established the different cases, it's time to dive into solving the equations you've set up.

• For the first equation where both expressions inside the absolute values are equal (\(2-x = 3x+2\)), move all terms to one side to isolate variables. Here, you'll find \(x = 0\).
• In the second scenario (\(2-x = -3x-2\)), the trick is to handle the negative expression properly. Rearrange and solve for \(x\), which results in \(x = -2\).

Each step involves basic algebraic principles: separating terms, combining like terms, and using multiplication or division as necessary to isolate the variable. By carefully aligning each term, you unveil potential solutions, solving for the elusive values of \(x\).

Checking Solutions

After solving, it's essential to ensure correctness by checking potential solutions against the original equation. This final step validates your results.

• Plug each solution back into the original equation \(|2-x| = |3x+2|\).
• For \(x = 0\), you substitute and confirm both sides equal \(2\), affirming it's a valid solution.
• For \(x = -2\), substitution shows both sides equal \(4\), verifying this solution too.

Checking solutions are like a concluding celebration where everything aligns perfectly—it ensures the reliability of your answers. In mathematics, this verification step is crucial as it guarantees the results are not just correctly derived but also applicable in the given equation. Think of it as giving your answers a stamp of approval.