Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-14,\) write the solution set of each equation. $$ |7-x|+2=12 $$

Step-by-Step Solution

Answer

The solution set is \(-3, 17\).

Step 1: Isolate the Absolute Value

Start by isolating the absolute value expression. We have \(|7-x| + 2 = 12\). First, subtract 2 from both sides of the equation to isolate the absolute value: \[|7-x| = 12 - 2\] This simplifies to: \[|7-x| = 10\]

Step 2: Solve for Positive Scenario

The absolute value \(|a| = b\) results in two possible cases: \(a = b\) or \(a = -b\). For the positive scenario, set the expression inside the absolute value equal to 10. \[ 7-x = 10 \] Solve for \(x\) by subtracting 7 from both sides: \[ -x = 10 - 7 \] This gives \(-x = 3\). Now, multiply by -1 or divide by -1 to solve for \(x\): \[ x = -3 \]

Step 3: Solve for Negative Scenario

For the negative scenario, set the expression inside the absolute value equal to -10. \[ 7-x = -10 \] Solve for \(x\) by subtracting 7 from both sides: \[ -x = -10 - 7 \] This simplifies to \(-x = -17\). Multiply by -1 or divide by -1 to solve for \(x\): \[ x = 17 \]

Step 4: Write the Solution Set

The solutions from the positive and negative scenarios are \(x = -3\) and \(x = 17\). Therefore, the solution set is: \[ \{-3, 17\} \]

Key Concepts

Solution Set

A solution set is a collection of all values that satisfy a given equation. When working with absolute value equations, such as \(|7-x| + 2 = 12\), the task is to find every possible value of \(x\) that makes the equation true. Once we've determined the possible values that \(x\) can take, we list these values in a set format, often using curly braces, like so: \( \{-3, 17\} \). This solution set indicates that both \(-3\) and \(17\) will balance the original equation, confirming their legitimacy as solutions through verification.Evaluating a solution set involves substituting each value back into the original equation to ensure that it holds true. If any value fails to satisfy the equation, it would not belong in the solution set. In this example, both solutions do satisfy the equation. Therefore, \(\{-3, 17\}\) is the complete solution set for the given absolute value equation.

Solving Equations

Solving equations, especially those with absolute values, involves finding values for the variables that satisfy the equation. This requires understanding that absolute values output non-negative numbers, forming two possible scenarios for solutions:

• The expression inside the absolute value equals the number itself (\(a = b\))
• The expression inside the absolute value equals the negative of the number (\(a = -b\))

Through logical steps and algebra rules, we can solve for \(x\) efficiently. For example, in \(|7-x| = 10\), one scenario is \(7-x = 10\) and the other is \(7-x = -10\). Solving these simpler equations provides potential solutions for \(x\). Each possible solution needs to be checked to ensure no extraneous solutions are included.

Isolation of Absolute Value

The first step when solving equations with absolute values is to isolate the absolute value expression. This makes it easier to work on the positive and negative scenarios that absolute values create. In \(|7-x| + 2 = 12\), start by isolating \(|7-x|\) through subtraction: \[|7-x| = 12 - 2 \]This simplifies to \(|7-x| = 10\). With the absolute value isolated, the work becomes straightforward. This approach ensures you only manipulate the absolute value, making the rest of the process more straightforward re-tractions directly on the potential solutions.

Positive and Negative Scenarios

When an absolute value equation is isolated, the next step is to consider the positive and negative scenarios for the expression inside the absolute value. For \(|a| = b\), this means:

• \(a = b\)
• \(a = -b\)

Taking the equation \(|7-x| = 10\) as an example, for the positive scenario we assume \(7-x = 10\) and solve for \(x\) to get \(-3\). For the negative, we assume \(7-x = -10\), leading to \(x = 17\). Both values are then verified against the initial equation.This dual-scenario approach is essential because it aligns with the nature of absolute values, which can move from both sides of zero. Understanding these scenarios ensures comprehensive solutions, eliminating the risk of missing any viable answers.