Problem 93 - Intermediate Algebra

Question

Solve each equation. $$ |3 x-7|+8=22 $$

Step-by-Step Solution

Answer

The solutions are \(x = 7\) and \(x = \frac{-7}{3}\).

Step 1: Isolate the Absolute Value Expression

To isolate the absolute value expression, subtract 8 from both sides of the equation:\[|3x - 7| + 8 - 8 = 22 - 8\]which simplifies to:\[|3x - 7| = 14\]

Step 2: Write Two Separate Equations

The equation \(|3x - 7| = 14\) means that the expression \(3x - 7\) can be either 14 or -14. Write two different equations to solve for \(x\):1. \(3x - 7 = 14\)2. \(3x - 7 = -14\)

Step 3: Solve the First Equation

Solve \(3x - 7 = 14\) by first adding 7 to both sides:\[3x = 14 + 7\]which simplifies to:\[3x = 21\]Then divide by 3:\[x = \frac{21}{3} = 7\]

Step 4: Solve the Second Equation

Solve \(3x - 7 = -14\) by first adding 7 to both sides:\[3x = -14 + 7\]which simplifies to:\[3x = -7\]Then divide by 3:\[x = \frac{-7}{3}\]

Step 5: Write the Solution Set

The solutions to the equation are the results of both solved equations. Therefore, the solution set is:\[x = 7 \quad \text{and} \quad x = \frac{-7}{3}\]

Key Concepts

Solving Equations

Equations, in Mathematics, are statements that assert the equality of two expressions. Solving equations involves finding the value of the variable that makes the equation true. When we talk about absolute value equations, it means we're dealing with expressions inside those vertical bars \(|...|\), which denote the distance of a number from zero on the number line regardless of direction.

The first step in solving any absolute value equation is to isolate the absolute value expression. This means we need to get it to one side of the equation by itself. In our example, \(|3x-7| + 8 = 22\), we achieve this by subtracting 8 from both sides, turning it into \(|3x-7| = 14\).

The fundamental concept here is to understand that an absolute value equation represents two different equations because the expression inside the absolute value can either be positive or negative—but both lead to a positive output. This dual-natured property comes in handy in finding all possible solutions for x.

Step-by-Step Solutions

Now, let's dive into solving the equation we derived, \(|3x - 7| = 14\). The key principle is to reconstruct two separate equations from the absolute value equation:

• 3x - 7 = 14
• 3x - 7 = -14

These are the two possibilities one should consider, seeing as the absolute value can result from a positive or negative version of the expression.

For the first equation, \(3x - 7 = 14\), simply add 7 to both sides to isolate \(3x\) and then divide by 3. This results in a neat solution, \(x = 7\).

Switching to the second equation, \(3x - 7 = -14\), apply the same operations: add 7 to both sides which simplifies to \(3x = -7\). Divide by 3 to find \(x = \frac{-7}{3}\). With methodical calculations, each value of x emerges smoothly and becomes part of the solution set.

Solution Set

The solution set for an equation is a collection of all values that satisfy the equation, making it true. For the equation \(|3x - 7| + 8 = 22\), after our operations, we determined two potential solutions for x: 7 and \(\frac{-7}{3}\).

These solutions represent the two scenarios of the equation emanating from the absolute value's operation on possible positive and negative expressions inside its scope.

In summary, the final solution set is: \{x = 7, x = \frac{-7}{3}\}\. This simple, comprehensive collection of solutions encapsulates all the x values that answer the original absolute value equation. Understanding solution sets is critical as they help to encapsulate and convey the summary of all possible solutions that satisfy the context of the problem.