Solve each equation or inequality. |8 - 3x| - 3 = -2

Verified step by step guidance

1. Start by isolating the absolute value expression on one side of the equation. Add 3 to both sides to get: \(|8 - 3x| = -2 + 3\).
2. Simplify the right side of the equation: \(|8 - 3x| = 1\).
3. Recall that the absolute value equation \(|A| = B\) means \(A = B\) or \(A = -B\), provided that \(B \geq 0\). Since \(B = 1\) here, set up two separate equations: \$8 - 3x = 1\( and \)8 - 3x = -1$.
4. Solve each equation for \(x\) separately. For \$8 - 3x = 1\(, subtract 8 from both sides and then divide by -3. For \)8 - 3x = -1$, do the same: subtract 8 and divide by -3.
5. Write the solutions from both equations as the solution set to the original equation.

Key Concepts

Absolute Value Equations

An absolute value equation involves expressions within absolute value bars, which represent the distance from zero on the number line. To solve, isolate the absolute value expression and then set up two separate equations: one where the expression equals the positive value, and one where it equals the negative value.

Isolating the Absolute Value Expression

Before solving an absolute value equation, you must isolate the absolute value term on one side of the equation. This often involves adding or subtracting constants and dividing or multiplying coefficients to simplify the equation for easier solving.

Solving Linear Equations

After splitting the absolute value equation into two linear equations, solve each by isolating the variable using inverse operations like addition, subtraction, multiplication, or division. This yields the possible solutions to the original equation.