True or False?

1. x-a is the difference between x and a.

2. If x - a>0 then x-a is the distance between x and a.

3. If x-a < 0 then x − a is the distance between x and a.

4. x=x-0

1. Reread the definitions above two or three times, then answer the following questions.

(a) Applying the definition of the absolute value, | − 5| is the distance between _______and________.

(b) Draw a curly brace to indicate the distance between -5 and zero on the real number line below.

Label the curly brace with the expression “| — 5|". (Which side of 0 is it on? Does this make sense?)

(c) What is the distance between -5 and zero?

(d) Similarly draw 5 on the same number line and label it, then complete the statements below.

By the definition of the absolute value |5| = ____________because the distance between _________and _________ is _____.

The absolute value - 5 = _________because the distance between _____________ and _______is ___________.

2. By definition, |x| = 6 means that the distance between ________ and 0 is __________.

On the number line, indicate all possible values of x that solve the equation |x| = 6.

3. We saw that |x| = 6 means the distance between x and 0 is 6. We also noted that |x| = 6 is equivalent to |x − 0| = 6.

(a) Interpret the meaning of |x − 4| = 6 and write this statement as a complete sentence.

(b) Mark all values that x that satisfy the sentence you wrote in (a) and explain your reasoning to your group.

(c) Another way to interpret |x − 4| = 6 is to notice that |x − 4] = 6 is equivalent to [(x − 4) − 0| = 6.Then we can say that the distance between (x − 4) and 0 is _________. Here is a picture

Solve for x:

In which graph above did you find these same values for x?

4. Solve the absolute value equations for x and graph your solutions on a number line. \text { (a) }|5 x-2|=13

\text { (c) } 3+|4 x-1|=8 5. Absolute Value Inequalities

(a) Use the definition of the absolute value to interpret the meaning of the absolute value inequality x < 6. Write your interpretation as a complete sentence. (Your sentence should start with, "The distance between...”)

(b) Label the tic marks for -10, -2,0, 2, 4, 8, 6 on the number line below. Determine which of these values satisfy the sentence that you wrote in part (a). Ask yourself:

Shade all values of x on the number line that satisfy the sentence you wrote in part (a)

(c) Use an inequality to describe the shaded region on the number line in (b).

(d) Notice that |x| < 6 is equivalent to |x − 0| < 6. Also notice that ___________ .is in the center of the shaded region above and there is a length of ___________ on either side of the center.

• Indicate on the number line in (b) where we see the center and length of _____________ on each side of the center.

• Draw arrows to indicate where these values are in the inequality |x − 0| < 6.

6. Interpret the inequality |x| ≥ 6 using the definition of the absolute value then shade this region on the number line.

Shade all values of x on the number line that satisfy the sentence you wrote in part (a)

(a) Write the region described by |x| ≥ 6 using a compound inequality: 7. Let's consider the inequality |x − 4| ≤ 5 using the definition of the absolute value.

(a) First, note that |x - 4 ≤ 5 is equivalent to [(x − 4) − 0| ≤ 5.Then |x − 4| ≤ 5 means that the distance between __________________ and 0 is less than or equal to ___________. We can illustrate this on the number line:

Write the compound inequality that describes the shaded region and solve for x.

Graph the solution set on the real number line:

(b) Look at the set of solutions for |x − 4| ≤ 5 that you found in the last exercise. What is the centre of this set? How long is the set on either side of the centre? Where do you see these in the statement of the inequality?

8. We can similarly solve the inequality |x − 4| ≥ 5.

The inequality |x - 4 ≤ 5 means that the distance between _______________ and 0 is greater than or equal to_____________. We can illustrate this on the number line: