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

Verified step by step guidance

1. Isolate the absolute value expression by subtracting 12 from both sides of the equation: -4|x+1| = -12.
2. Divide both sides of the equation by -4 to solve for the absolute value expression: |x+1| = 3.
3. Set up two separate equations to account for the absolute value: x+1 = 3 and x+1 = -3.
4. Solve each equation for x. For the first equation: x+1 = 3, subtract 1 from both sides to find x = 2. For the second equation: x+1 = -3, subtract 1 from both sides to find x = -4.
5. The solutions to the equation are x = 2 and x = -4.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x|, which equals x if x is non-negative and -x if x is negative. Understanding absolute value is crucial for solving equations that involve expressions within absolute value bars, as it leads to two possible cases to consider.

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. In the context of the given equation, once the absolute value is isolated, it can be transformed into linear equations that can be solved for the variable x.

Isolating the Variable

Isolating the variable is a fundamental algebraic technique used to solve equations. This involves rearranging the equation to get the variable on one side and the constants on the other. In the case of the given equation, isolating the absolute value expression is the first step to simplifying the problem and finding the solution for x.