Problem 22 - Precalculus Student Solutions Manual 5th

Question

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

Step-by-Step Solution

Answer

The solution is \x = 2\.

Step 1: Understand Absolute Value Equations

When solving equations involving absolute values, consider the definition of absolute value. If \(a\) and \(b\) are real numbers, then \|a| = |b|\ implies that \(a = b\) or \(-a = b\).

Step 2: Set Up the Equations

Given \|3 - 2x| = |5 - 2x|\, this can be broken down into two separate equations: \(3 - 2x = 5 - 2x\) and \|3 -2x| = |-(5 - 2x)|\ which simplifies to \3 - 2x = -(5 - 2x)\.

Step 3: Solve the First Equation

Solve \(3 - 2x = 5 - 2x\): Subtract \(3 - 2x\) from both sides to get: \(0 = 2\). This results in a contradiction, thus no solution from this equation.

Step 4: Solve the Second Equation

Solve \(3 - 2x = -(5 - 2x)\): Distribute the negative sign: \(3 - 2x = -5 + 2x\). Add \(2x\) to both sides: \(3 = -5 + 4x\). Add \5\ to both sides: \8 = 4x\. Divide both sides by \4\ to isolate \x\: \x = 2\.

Step 5: Verify the Solution

Substitute \(x = 2\) back into the original equation to check: \|3 - 2(2)| = |5 - 2(2)|\ results in \|3 - 4| = |5 - 4|\, which simplifies to \|1| = |1|\. Since both sides are equal, \x = 2\ is a valid solution.

Key Concepts

absolute value

The absolute value of a number refers to its distance from zero on a number line, regardless of direction. It is always a non-negative number. In mathematical terms, for any real number \(a\), the absolute value is denoted by \(|a|\). This brings us to an important property: if \(|a| = |b|\), then it means either \(a = b\) or \(a = -b\). By understanding this core concept, solving equations that involve absolute values becomes more manageable. It’s crucial because it tells us that every absolute value equation can be split into two cases – the positive and the negative scenarios. Hence, for \(|3 - 2x| = |5 - 2x|\), finding solutions involves considering both \(3 - 2x = 5 - 2x\) and \(3 - 2x = -(5 - 2x)\). The result from these cases will offer possible solutions which we later verify.

equations

An equation is a mathematical statement asserting that two expressions are equal. In our example, \(|3 - 2x| = |5 - 2x|\) is an absolute value equation. To solve this, we need to break it into simpler parts. First, we consider the scenario where the expressions inside the absolute values are equal: \(3 - 2x = 5 - 2x\). Solving this gives us a contradiction (0 = 2), meaning this case has no solutions. The second scenario involves the opposite: \(3 - 2x = -(5 - 2x)\). Distributing the negative sign and solving, we find that \(x = 2\). Solving equations step-by-step involves:

• Breaking down complex terms
• Isolating the variable
• Simplifying the expression

These systematic steps help us navigate through absolute value equations smoothly.

verification

Verification is a crucial step in problem-solving. It ensures that the solutions obtained are indeed correct. After solving \(|3 - 2x| = |5 - 2x|\) to get \(x = 2\), we must check if this value satisfies the original equation. Substitute \(x = 2\) back into the original: \(|3 - 2(2)| = |5 - 2(2)|\). This simplifies to \(|1| = |1|\), which holds true. Thus, \(x = 2\) is a verified and valid solution. Verification helps catch any algebraic errors and confirms that the solution fits all parts of the equation. Always remember:

• Substitute the solution back into the original equation
• Simplify to check if both sides are equal
• Ensure no steps were skipped during solving

Thus, verification is essential for complete and accurate problem-solving.