Solve: $6\mid 1\text{–}2x\mid \text{—}7=11$.

Verified step by step guidance

1. Start by isolating the absolute value expression. Add 7 to both sides of the equation: 6|1 – 2x| – 7 + 7 = 11 + 7, which simplifies to 6|1 – 2x| = 18.
2. Next, divide both sides of the equation by 6 to solve for the absolute value: |1 – 2x| = 18 / 6, which simplifies to |1 – 2x| = 3.
3. Recall that the absolute value equation |A| = B means A = B or A = -B. So, set up two separate equations: 1 – 2x = 3 and 1 – 2x = -3.
4. Solve each equation for x separately. For 1 – 2x = 3, subtract 1 from both sides and then divide by -2. For 1 – 2x = -3, subtract 1 from both sides and then divide by -2.
5. Write the two solutions for x obtained from the previous step. These are the values of x that satisfy 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 such equations, set the expression inside the absolute value equal to both the positive and negative values of the number on the other side of the equation.

Isolating the Absolute Value Expression

Before solving an absolute value equation, isolate the absolute value term on one side of the equation. This often involves performing inverse operations like addition, subtraction, multiplication, or division to simplify the equation and make the absolute value expression alone.

Solving Linear Equations

After splitting the absolute value equation into two cases, solve each resulting linear equation by isolating the variable. This includes applying inverse operations and simplifying to find the values of the variable that satisfy the original equation.