Absolute Value Equations

Front

What is the definition of absolute value?

Back

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted as |x|.

Front

How do you solve an absolute value equation of the form |x| = a?

Back

To solve |x| = a, you set up two equations: x = a and x = -a.

Front

What is the first step in solving an absolute value equation?

Back

The first step is to isolate the absolute value expression on one side of the equation.

Front

What does the equation |2x - 3| = 4x - 1 imply about the values of x?

Back

It implies that the expression inside the absolute value can equal either 4x - 1 or -(4x - 1), leading to two separate equations to solve.

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.

Front

What are the possible solutions for the equation |5x| = 40?

Back

The possible solutions are x = 8 and x = -8.

Front

What is the significance of the equation |x - 1| = 5x + 10?

Back

This equation indicates that the expression inside the absolute value must be equal to 5x + 10 or its negative, leading to two equations.

Front

How do you handle absolute value equations that result in a contradiction?

Back

If solving leads to a contradiction (e.g., a false statement), it means there are no solutions.

Front

What is the solution set for the equation |2x - 1| + 4 = 8?

Back

The solution set is x = 2.5 and x = -1.5.

Front

What is the general approach to solving equations of the form |ax + b| = c?

Back

Isolate the absolute value, then set up two equations: ax + b = c and ax + b = -c.