Download Help Examples for w4 1. The equation |3x−18| = 9 has two solutions

Help Examples for w4
1. The equation |3x−18| = 9 has two solutions. Find the sum of those two solutions.
(Notice that it is asking you to find the solutions and then add the two solutions as
your answer.)
Solution: Taking off the absolute sign, the equation becomes 3x − 18 = 9 and
3x − 18 = −9. Solving the first gives x = 9 and solving the second gives x = 3. The
sum of the two solutions is thus 9 + 3 = 12.
2. The equation |4x + 8| = 16 has two solutions. Find the distance between those
two solutions.
This problem is similar to the first one, except that after you find the two solutions,
you need to find the “distance” between them, which is obtained by taking the larger
solution minus the smaller solution (the distance has to be a positive number).
3. Solve the inequality |2x − 4| ≤ 26. Write the answer in interval notation. Note:
If the answer includes more than one interval write the intervals separated by the ”union”
symbol, U. If needed enter ∞ as infinity and −∞ as -infinity .
Solution. Step one: Change the inequality into an equation: |2x − 4| = 26. Solve
this equation: 2x − 4 = ±26 which yields x = 15 and x = 11.
Step two: The two solutions divide the real number line (namely the entire interval
(−∞, ∞)) into three parts: (−∞, 11], [11, 15] and [15, ∞) (square brackets are used on
the numbers 11 and 15 since they are solutions (the inequality included the = case).
Step 3: Choose a number from an interval to test. If you take 0, for example, plug it
in the inequality for x: |2 · 0 − 4| ≤ 26 becomes 4 ≥ 26, which is not true. So 0 is not a
solution. Since 0 is in the leftmost interval (−∞, 11], that means (−∞, 11] is not in our
solution set. By the general alternating rule, the interval adjacent to it (namely [11, 15])
is in the solution set and the next one is not. Thus the solution set is just [11, 15].
Problems 4 to 6 are similar to #3. Just be careful when you solve the equation and
carry out the test procedure.
7. It is again similar to #3. The only difference is that when you get your answer
as an interval such as [−2, 5], then the smaller number (−2 in my example) is entered
as the number A and the bigger number (5 in my example) is entered as B. You also
need to determine which intervals given is in the solution set.
8. Express the inequality using interval notation. −5 < x ≤ 5.
Solution. (−5, 5] (If you cannot even do this, then you were totally wasting your
time in class since you apparently were not paying any attention.)
10. (1 pt) Solve the equation.
1
|3−2x|
=3
Solution. Taking reciprocal on both sides, the equation is simply |3−2x| = 1/3 so you
get 3 − 2x = 1/3 and 3 − 2x = −1/3. Multiply both sides by 3 (the only denominator)
we get 9 − 6x = 1 and 9 − 6x = −1 so x = 8/6 = 4/3 and x = 10/6 = 5/3. You are
asked to enter the answers as two numbers separated by a comma: 4/3, 5/3
#12–#14: note that these inequalities do not have absolute signs so when you solve
the equations, there is only one solution to each equation.
15. Solve the inequality : |3x − 6| > 2. The solution of this inequality consists of one
or more of the intervals (−∞, A), (A, B) and (B, ∞) where A < B. For each interval,
answer YES or NO to whether the interval is included in the solution.
Solution. This is just like #7 which is solved like #3. Let us go over this one more
time:
Step one: Change the inequality into an equation: |3x − 6| = 2. Solve this equation:
3x − 6 = ±2 which yields x = 4/3 and x = 8/3.
Step two: The two solutions divide the real number line (namely the entire interval
(−∞, ∞)) into three parts: (−∞, 4/3), (4/3, 8/3) and (8/3, ∞) (parentheses are used
on the numbers 4/3 and 8/3 since they are not solutions of the inequality (the inequality
does not include the = case).
Step 3: Choose a number from an interval to test. If you take 0, for example, plug
it in the inequality for x: |3 · 0 − 6| > 2 becomes 6 > 2, which is true. So 0 is a solution.
Since 0 is in the leftmost interval (−∞, 4/3), that means (−∞, 4/3) is in our solution
set. By the general alternating rule, the interval adjacent to it (namely (4/3, 8/3)) is
not in the solution set and the next one (namely (8/3, ∞)) is. Thus the solution set is
(−∞, 4/3)∪(8/3, ∞). For this particular question, that means you should enter A = 4/3
and B = 8/3 as your answer. If you were asked to enter the intervals as your answer,
you should simply type in (−inf inity, 4/3)U (8/3, inf inity).
16. Similar to #15.
17. Each question corresponds one and only one answer. So match them up carefully.
#18–#21: You have seen each kind of equation or inequality already!