Solving Absolute Value Equations

Front

What is an absolute value equation?

Back

An absolute value equation is an equation that contains an absolute value expression, which represents the distance of a number from zero on the number line.

Front

How do you solve an absolute value equation?

Back

To solve an absolute value equation, isolate the absolute value expression and then set up two separate equations: one for the positive case and one for the negative case.

Front

What does the equation |x| = a imply?

Back

The equation |x| = a implies that x can be either a or -a, provided that a is non-negative.

Front

What is the first step in solving the equation |4x - 2| + 10 = 16?

Back

The first step is to isolate the absolute value expression by subtracting 10 from both sides, resulting in |4x - 2| = 6.

Front

What are the two cases to consider when solving |x - 1| + 2 = 10?

Back

The two cases are: 1) x - 1 = 8 (positive case) and 2) x - 1 = -8 (negative case).

Front

What does it mean if an absolute value equation has no solution?

Back

It means that there are no values of the variable that can satisfy the equation, often occurring when the absolute value expression is set equal to a negative number.

Front

How do you interpret the solution set of an absolute value equation?

Back

The solution set consists of all values of the variable that satisfy the original equation, which can include one, two, or no solutions.

Front

What is the significance of the absolute value symbol in equations?

Back

The absolute value symbol indicates that the expression inside can be either positive or negative, reflecting the concept of distance from zero.

Front

What is the solution to the equation |2x + 9| = -13?

Back

There is no solution because the absolute value cannot be negative.

Front

How do you check the solutions of an absolute value equation?

Back

Substitute the solutions back into the original equation to verify if both sides are equal.