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, and solve for x.

Front

What is the first step in solving an absolute value equation like |2x - 3| + 5 = 10?

Back

Subtract 5 from both sides to isolate the absolute value expression: |2x - 3| = 5.

Front

What are the possible cases when solving |x - a| = b?

Back

The two cases are: x - a = b and x - a = -b.

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 because the expression inside the absolute value cannot equal a negative number.

Front

How do you interpret the solution set {−1, 4} for the equation |2x - 3| + 5 = 10?

Back

It means that both x = -1 and x = 4 are valid solutions to the equation.

Front

What is the significance of the equation |−2n| + 10 = −50 having no solution?

Back

It indicates that the left side, being an absolute value plus a positive number, can never equal a negative number.

Front

How do you check if your solutions to an absolute value equation are correct?

Back

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

Front

What is the general form of an absolute value equation?

Back

The general form is |expression| = constant, where the expression can be any algebraic expression.

Front

What is the solution to the equation |x - 12| - 3 = 11?

Back

The solutions are x = -2 and x = 26.