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

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

Back

An absolute value equation has no solution if the expression inside the absolute value cannot equal the negative value on the other side of the equation.

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, provided a is non-negative.

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

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

Back

If the absolute value expression equals a negative number, the equation has no solution.

Front

What is the solution set for the equation |x + 3| = 5?

Back

The solution set is {2, -8}, derived from the equations x + 3 = 5 and x + 3 = -5.

Front

What is the significance of the equation |x| + c = d?

Back

For |x| + c = d, the equation has solutions if d is greater than or equal to c; otherwise, it has no solution.

Front

How do you interpret the equation |x - 4| = 6?

Back

This equation means that the distance between x and 4 is 6, leading to two possible solutions: x = 10 and x = -2.

Front

What is the process for solving |ax + b| = c?

Back

First, isolate the absolute value, then set up two equations: ax + b = c and ax + b = -c, and solve for x.

Front

What does the equation |v + 8| - 14 = -7 imply?

Back

This equation implies that |v + 8| = 7, leading to two solutions: v + 8 = 7 and v + 8 = -7.