Matrices & Inequalities Review Sheet ANSWERS
1. n = 16
2
7
2. 

 6  7 
3. w = –37
x = 18
z = –17
y=4
4.
–4
5. x ≥ 0
6. x  1  3
7. x < 19
19
8. y < –1
–1
9. x > 4
OR x < –7
–7
4
10. x ≥ –8/3 AND x ≤ 4
–8/3
4
11. NO SOLUTION
12. All Real Numbers
13. x  170  10
x ≤ 180 AND x ≥ 160
14. $3.75 + $2.99c ≤ 20
c ≤ 5.43 (meaning he can buy 5 bags or less)
15. k > 3
OR
k < –7
(open circles)
–7
3
16. t  28  1.2
t ≤ 29.2 AND t ≥ 26.8
17. A  832  46
18.
3
h  3  12
4
h ≥ 12
19. All Real Numbers
20. No solution
21.
–2
22. –5x < –6x + 12
Solution set: x < 12
23. 1.50a – 0.30a ≥ 75
24. x = 32
y = –9
25. a = 49
b=
 35
3
c = –5
26.
9
2
2
-10
3
1
3
7
27.
28.
29. x  80  5
30. From –4 degrees to 6 degrees.
31.
11
32. Slowest: 24.94
19
Fastest: 25.06
d = 16
NOTES ON THE PROBLEMS:
1. Find your scalar by setting up an equation between 2 of the numbers that you know in both matrices. For
example, –6x = –8, where x is your unknown scalar. By solving this equation, you see that your scalar (x) is
4/3. Now, multiply that scalar by 12, and you see that n = 16.
2. If you “keep, change, opposite,” be sure to “opposite” every number in matrix B.
3. One way of doing this problem is to set up four equations involving corresponding elements and solve them.
For example, –23 – 3w = 88.
4. Be sure you “flip” your sign when you multiply or divide by a negative (yes, –1 is in front of x). ALSO,
before you graph, get your x on the left!!!!
5. It is very much okay to have 0 as the number. Treat this problem just as if you had any other number on the
right side of the inequality.
6. Find the middle number in between –2 and 4 by averaging the two (add and divide by 2). The difference
(subtraction) of any “jump” and your middle number is what goes inside absolute value. The distance from
your middle number to your endpoints is what the absolute value is less than (less than because the number line
is shaded on the inside).
7. Remember, 2x – 3 is “stuck” over the 5, so before you can do anything to get rid of 2x – 3, you must get rid
of the 5 first.
8. Be careful on distributing the last part of the problem (signs)!!! –2 (–1 – y)
9. Set up 2 problems. Be sure to flip your sign on the second problem.
10. Set up 2 problems. Be sure to flip your sign on the second problem.
11. Get rid of the “visitor” (–3). Don’t forget to flip the sign when multiplying or dividing by a negative!!!
LOOK FOR THE RED FLAG after the “visitor” is gone.
12. Get rid of all “visitors” (– 12 first then –2 in front). Don’t forget to flip the sign when multiplying or
dividing by a negative!!! LOOK FOR THE RED FLAG after the “visitor” is gone.
13. Takeoff has to be “within” 10…meaning 10 above 170 or 10 below 170 OR anything in between.
14. Do not get this confused with an absolute value word problem!!! This has nothing to do with absolute
value.
15. Solve the equation they give you. Be sure to make your two problems and flip your sign on the second
problem.
16. My thermometer is “within” 1.2 degrees, meaning it could be 1.2 above OR 1.2 below OR anything in
between.
17. 832 is what I am aiming for. The distance Amanda’s guess (A) is from 832 (subtraction problem inside
absolute value) is “within” 46 pieces. She could be 46 above, 46 below, or anything in between.
18. Three-fourths OF means multiply. She has already practiced three hours so add that back to her time.
19. Your c’s will cancel on both sides and leave you with 6 ≥ 6, which is ALWAYS true.
20. Once you distribute and combine like terms on the right side, your variables disappear and you are left with
46 ≤ –20, which is not true.
21. Be sure to flip the inequality sign if you divide by a negative. Before you graph the answers on a number
line, get the x on the left.
22. “Is less than” means <. Sum means add.
23. AT least means she wants to make $75 or more, which is ≥ 75. When she sells the apples, she gets money.
We have to subtract $0.30 per apple ($0.30a) because it is taking away from her money she is making since it
“costs” her.
24. Set up the equations and solve. BE SURE TO DISTRIBUTE –1 TO EVERYTHING IN THE SECOND
MATRIX!. Two equations: (4x – 9) – (x – 11) = 98
and
10 – (3y + 5) = y + 41
25. Distribute the scalars (3 and –2) to both matrices, set up your four equations, and solve
Four equations: 84 – 2a = –14
–36 – 2b = b – 1
15 – 2c = 25
3d + 27 – 2d = 43
26. No “visitors” so make your two problems (switch the sign on the second) and solve. The original sign is
greatOR, so the number line should be shaded “out.”
27. Get rid of the “visitor” (–3) by dividing. BE SURE TO FLIP YOUR SIGN SINCE YOU DIVIDED BY A
–3! Make your two problems (switch the sign on the second) and solve. The original sign (after the visitors are
gone) is less thAND, so the number line should be shaded “in.”
28. Get rid of the “visitor” (– 12) by adding 12. Make your two problems (switch the sign on the second) and
solve. The original sign (after the visitors are gone) is greatOR, so the number line should be shaded “out.”
29. The perfect (middle) temperature is 80, so it goes inside the absolute value. The distance I can be away
from 80 is 5 degrees, so it goes at the end of the problem. I am between (inside) that distance of 5, so the sign
should be less than or equal to.
30. Perfect temp is 1 degree. I can be 5 degrees above (1 + 5) or 5 degrees below (1 – 5) that perfect temp.
31. Marcus can work 4 hours less than 15 (15 – 4 = 11) or 4 hours more than 15 (15 + 4 = 19).
32. Change the percent to a decimal (0.06), then add it to and subtract it from 25.