In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. 2|4 - (5/2)x| + 6 = 18

Verified step by step guidance

1. Start by isolating the absolute value expression. Subtract 6 from both sides of the equation: 2\left|4 - \frac{5}{2}x\right| + 6 - 6 = 18 - 6, which simplifies to 2\left|4 - \frac{5}{2}x\right| = 12.
2. Next, divide both sides of the equation by 2 to further isolate the absolute value: \frac{2\left|4 - \frac{5}{2}x\right|}{2} = \frac{12}{2}, giving \left|4 - \frac{5}{2}x\right| = 6.
3. Recall that if \left|A\right| = B, where B > 0, then A = B or A = -B. Apply this property to get two separate equations: 4 - \frac{5}{2}x = 6 and 4 - \frac{5}{2}x = -6.
4. Solve each equation for x separately. For the first equation, subtract 4 from both sides and then multiply both sides by the reciprocal of \frac{5}{2} to isolate x. Repeat the process for the second equation.
5. Check your solutions by substituting them back into the original equation to ensure they satisfy the absolute value 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 expression on one side of the equation. This often involves performing inverse operations like subtraction or division to simplify the equation and prepare it for splitting into two cases.

Solving Linear Equations

After splitting the absolute value equation into two linear equations, solve each one by applying algebraic techniques such as distributing, combining like terms, and isolating the variable. Check each solution in the original equation to verify its validity.