foundation and Structure
four less than three times m
3m - 4
4(x - 3) + 5 = 3x + 11
x = 18
Solve
F = (9/5)C + 32
for C.
C = (5/9)(F - 32)
Rewrite the ratio 5/6 : 3/4 using least whole numbers.
10:9
2 − [3 − (−2)³] + |−7 + 5|.
−7
(3x - 2)/4 - (x + 1)/6 = 5/12
x = 13/7
Solve V = lwh for h. State the restriction.
h = V/(lw)
, with lw ≠ 0.
Store A sells 1.8 L for $5.04. Store B sells 2.5 L for $6.75. Which is cheaper per liter, and by how much?
Store B. A is $2.80/L and B is $2.70/L, so B is cheaper by $0.10/L
Simplify and name the smallest real-number set: (√50 + √8)/√2.
7; natural number
3(2x - 5) - 4(x + 1) = 2x - 19
all real numbers.
Solve
q = (m + n)/(m − n)
for n.
n = m(q − 1)/(q + 1)
, with q ≠ −1; the original denominator must also be nonzero.
Convert 22 meters per second to kilometers per hour.
79.2 km/h
Evaluate efficiently without calculating the four products separately:
27(48)+27(52)+73(48)+73(52)
27(48+52)+73(48+52)=(27+73)(48+52)
=100×100
=10,000
For k(x − 1) + 2 = 5x − 3, classify the solutions for k = 5 and k ≠ 5.
k = 5: all real numbers. k ≠ 5: unique solution x = 1
Solve |2x − 3| = x + 6.
x = −1 or x = 9
A 5.4 m by 2.4 m wall uses 30 cm by 40 cm panels. Include 10% extra. Panels come in boxes of 10. How many boxes are needed?
108 panels fit the wall; 10% extra gives 118.8, so buy 119 panels or 12 boxes.
Define the operation a ◇ b = a + b − 4. Find its identity and the inverse of a.
Identity 4; inverse 8 − a
a(2x−1)+b=(a+3)x+7.
Find all ordered pairs (a,b)(a,b) for which the equation has:
a. infinitely many solutions;
b. no solution;
a. (a,b)=(3,10)
b. a = 3, b is not equal 10
The weighted mean is
T = (m₁t₁ + m₂t₂)/(m₁ + m₂)
. Solve for m₂.
m₂ = m₁(t₁ − T)/(T − t₂)
, with T ≠ t₂ for a unique value.
A vehicle travels half of a route at 12 km/h and the other half at 18 km/h. Find its average speed for the whole route.
14.4 km/h