Solve each equation in Exercises 96–102 by the method of your choice. -4|x+1| + 12 = 0

Verified step by step guidance

1. Rewrite the equation to isolate the absolute value expression. Start by subtracting 12 from both sides: -4|x+1| = -12.
2. Divide both sides of the equation by -4 to further isolate the absolute value: |x+1| = 3.
3. Recall the definition of absolute value: |A| = B implies two cases: A = B or A = -B. Apply this to the equation: x+1 = 3 or x+1 = -3.
4. Solve each case separately. For the first case, subtract 1 from both sides: x = 3 - 1. For the second case, subtract 1 from both sides: x = -3 - 1.
5. Write the solutions as a set of values for x. These are the solutions to the equation.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. Understanding how to manipulate absolute values is crucial for solving equations that involve them, as it often leads to two separate cases based on the definition of absolute value.

Linear Equations

A linear equation is an equation of the first degree, meaning it involves variables raised only to the power of one. The general form is ax + b = c, where a, b, and c are constants. Solving linear equations often involves isolating the variable on one side, which is essential when dealing with equations that arise from absolute value expressions.

Equation Solving Techniques

Various techniques can be employed to solve equations, including substitution, elimination, and graphical methods. In the context of absolute value equations, it is common to isolate the absolute value expression first and then set up two separate equations to account for both the positive and negative scenarios. Mastery of these techniques is vital for effectively finding solutions to complex equations.