Practice Problems (Ch. 2) M101
(1) Solve the equation:
(a) 2x - 3 = 5,
(b) 3(x - 4) = 4(x + 2) + 7,
(c)
(d)
(2) The sum of two consecutive integers is 85. Find the two integers (set the problem up as a linear equation, and solve - no credit will be given for guessing).
(3) A 14 ft. board is cut into three pieces. The middle piece is 2 ft. longer than the shortest piece. The longest piece is twice as long as the shortest piece. Find the length of each piece.
(5) (a) The perimeter of a rectangle is 22 inches. The length is 4 inches more than the. Find the width.
(b) Two boats sit 200 miles apart on a river. The northmost boat leaves traveling south at an average speed of 15 mph. The other boat heads North up river at 12 mph. How long will it be before the boats pass each other?
(c) Joe has $16,000 to invest in two funds, one paying interest at a rate of 4%, and the other paying 6%. If altogether, he earns $780 in interest, how much did he earn at each rate?
(6) Solve the inequality and graph the solution set:
(a)
(b)
(c)
(d)
Solutions:
(1) Of course, the strategy here is to isolate the variable. There may be different obstacles along the way, but (roughly) we clear fractions or decimals, distribute then combine like terms, add or subtract, then multiply or divide on both sides by the same number in order to get the variable by itself:
(a)
2x - 3 = 5 (Isolate x - first, get rid of the -3 by adding: )
+3 +3
2x = 8 (Now we get rid of the 2 by dividing)
(b) 3(x - 4) = 4(x + 2) + 7, (parentheses first - distribute)
3x - 12 = 4x + 8 + 7 (combine like terms 8 and 7)
3x - 12 = 4x + 15, (move variable to one side)
-3x -3x
-12 = x + 15 (get rid of the 15 - subract)
-15 -15
-27 = x
(c)
(clear fractions by multiplying through by the LCD = 6)
4x - 3 = 5, (now, isolate x)
+3 +3
4x = 8,
(d) (clear decimals by *'ing by 100 - 2 decimal places)
-40 -40
2x = 80,
x = 80/2,
x = 40.
(2)
identify unknowns:
n = first number,
n + 1 = second number (since they are consecutive)
translate information:
n + (n + 1) = 85 (combine and solve)
2n + 1 = 85,
-1 -1
2n = 84,
(3) You may want to draw a little picture here:
| | |
x x + 2 2x
The piece lengths have to add up to 14, so:
x + x + 2 + 2x = 14,
4x + 2 = 14
-2 -2
4x = 12,
(5) (a) We need to know the formula for the perimeter of a rectangle for this one. It is
P = 2l + 2w. We're given the perimeter P = 22, and the length l = 4 inches.
22 = 2(w+ 4) + 2w, (Isolate w:)
22 = 2w + 8 + 2w
-8 -8
14 = 4w, w = 14/4 = 7/2.
(b) We need to use the relationship between time, distance and rate:
d = rt,
25t
14t
These distances have to add up to 200:
25t + 14t = 200,
29t = 200,
(c) Let x = amount invested at 6%, and set up this table:
account 1 | x | .06 | .06x |
account 2 | 15,000 - x | .04 | .04(16,000 - x) |
| 780 |
This gives the equation:
(6) To solve an inequality like this, we need to isolate the variable, keeping in mind the 'special rule' for inequalities - if you multiply or divide by a negative number, you have to reverse the inequality.
(a)
+3 +3
2x < 10,
(b)
+3 +3
-2x < 10, (the special rule is in play, since we divide by -2:)
(reverse the inequality)
(c) (distribute first)
3x - 12 < 4x + 8 (move the variable to one side: )
-3x -3x
-12 < x + 8
-8 -8
-20 < x, same as: x > -20
(d) (Isolate on all sides at once)
-2 -2 -2
-6 < 2x < 2,
(b) Clear fractions first: