Problem 13 - Algebra 2

Question

Solve each equation. Check your solutions. \(|c-2|=2 c-10\)

Step-by-Step Solution

Answer

The solution is \(c = 8\).

Step 1: Understand the absolute value equation

The equation given is \(|c - 2| = 2c - 10\). We need to consider the property of absolute values, where \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\). This means we have two potential equations to solve based on the expression inside the absolute value.

Step 2: Solve for first case (positive case)

For the case where the expression inside the absolute value is non-negative, set \(c - 2 = 2c - 10\). Solving for \(c\): 1. Subtract \(2c\) from both sides: \(c - 2c - 2 = -10\)2. Simplify to find \(-c - 2 = -10\)3. Add 2 to both sides: \(-c = -8\)4. Multiply by -1: \(c = 8\).

Step 3: Solve for second case (negative case)

Consider the situation where \((c - 2) < 0\), meaning \(|c - 2| = -(c - 2)\). In this case, solve \(-(c - 2) = 2c - 10\): 1. Distribute the negative: \(-c + 2 = 2c - 10\)2. Add \(c\) to both sides to get: \(2 = 3c - 10\)3. Add 10 to both sides: \(12 = 3c\)4. Divide by 3: \(c = 4\).

Step 4: Check solutions

To verify, substitute back into the original equation:1. For \(c = 8\): - Left side: \(|8 - 2| = |6| = 6\) - Right side: \(2(8) - 10 = 16 - 10 = 6\) - Both sides are equal, so \(c = 8\) is a valid solution.2. For \(c = 4\): - Left side: \(|4 - 2| = |2| = 2\) - Right side: \(2(4) - 10 = 8 - 10 = -2\) - Both sides are not equal; thus, \(c = 4\) is not a valid solution.

Step 5: Conclusion

The only valid solution is \(c = 8\), as it satisfies the original equation in both cases checked.

Key Concepts

Checking Solutions

When solving equations, especially with absolute values, it's essential to verify solutions to ensure they satisfy the original equation. Let's break down why this step is important and how it can be done effectively. After finding potential solutions, always substitute them back into the original equation. This substitution helps in checking whether both sides of the equation remain equal.

• For example, if you have found a solution where \(c = 8\), plug it back into the original absolute value equation \(|c - 2| = 2c - 10\).
• Calculate both sides: Make sure to simplify both the absolute value side and the other side separately.
• Only consider solutions where both sides equal after substitution as valid.

Checking solutions is a safety measure that confirms accuracy. It prevents mistakes that could have arisen from algebraic manipulations during the solving process.

Step-by-Step Problem Solving in Algebra

Approaching algebra problems with a step-by-step method ensures thoroughness and accuracy. This method involves being systematic in handling and manipulating algebraic expressions. The original solution of \(|c - 2| = 2c - 10\), showcases this well:

• First, analyze the original equation: Determine the nature of expressions involved, like absolute values.
• Break it down: Split the problem into manageable steps, dealing first with each scenario of the absolute value.
• Solve each equation separately: As in the example, address the positive and negative scenarios one after the other.
• Proceed systematically: Make one change at a time, such as simplifying terms or rearranging the equation.

This step-by-step process aids in understanding and solving complex equations without rushing and missing crucial details.

Equation Verification

Equation verification is a crucial step to ensure that the solutions obtained satisfy the given equation. Once solutions are calculated, verifying involves substituting them back into the original equation to check for equality.

• Substitute explicitly: Take each potential solution and substitute it back into both sides of the original equation.
• Observe carefully: Simplify each side and see if they are equal.
• Discard invalid solutions: If the sides do not match, reject the solution as invalid, as evident in the solution where \(c = 4\) did not hold.

Verification ensures reliability and avoids accepting incorrect solutions which could invalidate your results. Consider it as a confirmation step.

Positive and Negative Case Analysis

Positive and negative case analysis is a method used to solve absolute value equations. These equations often have two scenarios due to the double nature of absolute values - they can be positive or negative. Here’s how this analysis works:

• Understand absolute value properties: The absolute value of a number changes if the number is less than zero.
• Set up different cases: Write equations for both \(x \geq 0\) and \(x < 0\).
• Solve each separately: Deal with them independently, as shown in the solution of \(c - 2 = 2c - 10\) and \(-(c - 2) = 2c - 10\).
• Analyze solutions: After solving for each case, determine which, if any, satisfy the original equation.

Conducting a positive and negative case analysis maximizes finding all potential solutions and helps distinguish valid from invalid results.