Solve each equation or inequality.
$\mid 7/2-3x\mid -9=0$

Verified step by step guidance

1. Start by isolating the absolute value expression on one side of the equation. Add 9 to both sides to get: \(|\frac{7}{2} - 3x| = 9\).
2. Recall that if \(|A| = B\), where \(B \geq 0\), then \(A = B\) or \(A = -B\). Apply this property to write two separate equations: \(\frac{7}{2} - 3x = 9\) and \(\frac{7}{2} - 3x = -9\).
3. Solve the first equation \(\frac{7}{2} - 3x = 9\) by isolating \(x\). Subtract \(\frac{7}{2}\) from both sides, then divide by \(-3\) to find \(x\).
4. Solve the second equation \(\frac{7}{2} - 3x = -9\) similarly: subtract \(\frac{7}{2}\) from both sides, then divide by \(-3\) to find \(x\).
5. Write the solution set as the two values of \(x\) found from the two equations.

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 Variable

Before solving, it is essential to isolate the absolute value expression on one side of the equation. This often involves adding or subtracting terms and dividing or multiplying both sides by constants to simplify the equation and make the absolute value term the subject.

Solving Linear Equations

After splitting the absolute value equation into two linear equations, solve each for the variable by performing inverse operations such as addition, subtraction, multiplication, or division. Check solutions in the original equation to ensure they are valid, especially when dealing with absolute values.