Intermediate Algebra Question

Question

The written expression, "The distance between x and a" can be translated as |x-a|. Consider the statement, "The distance between x and 5 is 3."

A) Think about a number line and find the two numbers that satisfy the statement.

B) Translate the statement into an equation and solve the equation. Show work.

Answer 1

A. On the number line, mark the number 5. Then to get to a number that is 3 units away from 5 count the units to the right first. This will lead to number 8. This is the first number which is 3 units away from number 5.

Next, count 3 units to the left of number 5. This will lead to number 2. This the second number which is 3 units away from number 5.

So the answers are x = 8 or 2 .

B. Follow the guidance in the first sentence of the problem. Distance between x and 5 can be written as |x-5|. This distance is equal to 3.

The equation can therefore, be written as

|x-5| = 3.

To solve this equation, you must realize that absolute value is always positive, even though the input values can either be negative or positive. So the absolute value equation we wrote above is actually equivalent to two distinct equations.

a) (x-5) is positive 3; the absolute value is positive 3 x-5 = 3

b) (x-5) is negative 3; its absolute value is positive 3 x-5 = -3

You can solve these two equations to find the two values of x. They will be the same answers as in A, above.