Plug It In
Evaluate 3x − 8 when x = 5.
Evalúa 3x − 8 cuando x = 5.
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Which property lets you find k in 3/5 + k = 9 + 3/5? And what is k? / ¿Qué propiedad permite hallar k? ¿Cuánto vale k?
Commutative Property; k = 9
The tell is that the order of the two terms swapped.
Solve −21 = p − 6. / Resuelve −21 = p − 6.
p = −15 — Watch for: −27, from subtracting 6 instead of adding it. Also 15, from losing the negative on the answer.
Solve |x| = 7. / Resuelve |x| = 7.
x = 7 or x = −7 — Watch for: 7 alone. This is the habit the rest of the category depends on.
Divide the seconds between lightning and thunder by 5 to get miles. The delay is 7.5 seconds. How far away? / El retraso es de 7.5 segundos. ¿A qué distancia está?
1.5 miles — Watch for: 37.5, from multiplying instead of dividing. Have them predict whether the answer should be bigger or smaller than 7.5 before computing.
If g = 2 and h = −4, find |h − g|.
Si g = 2 y h = −4, halla |h − g|.
If 8 + 22 = 30, then 30 = 8 + 22. Name the property of equality. / Nombra la propiedad de igualdad.
Symmetric Property of Equality
Watch for: Reflexive, which is a = a. Nothing here repeats itself; the two sides traded places.
Solve 4(x + 9) = 6x − 4. / Resuelve 4(x + 9) = 6x − 4.
x = 20 — Watch for: 6.5, from distributing to x but not to 9 (4x + 9). Also 3.2, from a sign error when collecting x-terms.
Solve |x + 5| = 2. / Resuelve |x + 5| = 2.
{−7, −3} — Watch for: −3 alone, from writing only x + 5 = 2. Also {7, 3}, from adding 5 instead of subtracting.
7 identical notebooks weigh 18 pounds total. The calculator shows 2.5714286. Give a reasonable estimate for one notebook. / Da una estimación razonable para un cuaderno.
About 2.6 pounds — Watch for: 2.5, from truncating rather than rounding. Also 3, which ignores "as a decimal," and copying the full 2.5714286.
If f = 3, g = −2, and h = −4, find 2|h − g| − |f − 6|.
Si f = 3, g = −2 y h = −4, halla 2|h − g| − |f − 6|.
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(2 · 2 − 3)
n + 0 = n. Name the property. / n + 0 = n. Nombra la propiedad.
Additive Identity Property
Watch for: Additive Inverse, which sums to zero rather than to n. Also the Zero Multiplicative Property, which belongs to multiplication.
Solve 4x − 5 = 9x + 45. / Resuelve 4x − 5 = 9x + 45.
x = −10 — Watch for: 10, with the sign dropped at the last division. Also stopping at −50 = 5x. Also 8, from moving 9x to the left as +9x instead of subtracting it.
Solve 3|y + 6| = 30. / Resuelve 3|y + 6| = 30.
{−16, 4} — Watch for: {−12, 8}, from subtracting 3 instead of dividing. Also {−4, 4}, from solving |y| = 10 and never undoing the +6.
Solve
(3x+15)/4=6y
for x. / Resuelve (3x + 15)/4 = 6y para x.
x = 8y − 5 — Watch for: 24y − 15, from multiplying by 4 and never dividing by 3. Also 8y − 15, from dividing only the 24y by 3.
Evaluate 5 − 2(a + 4) when a = −3.
Evalúa 5 − 2(a + 4) cuando a = −3.
3
Watch for: 15, from distributing −2 to a only and leaving +4 alone. Also 3(a + 4) from subtracting 5 − 2 first.
If x = 12 and 12 = y, then x = y. Name the property of equality. / Nombra la propiedad de igualdad.
Transitive Property of Equality
Watch for: Symmetric, because "12 = y" looks flipped. The giveaway is the chain through a shared middle value.
Is 7x + 5 = 7(x + 5) an identity, one solution, or no solution? Why? / ¿Es identidad, una solución o sin solución? ¿Por qué?
No solution, since 7x + 5 = 7x + 35 gives 5 = 35 — Watch for: Identity, chosen because both sides begin with 7x. Also "x = 0," confusing no solution with the solution being zero.
A number line shows closed dots at −3 and 7. Which equation matches: |x − 2| = 5, |x − 5| = 2, or |x + 2| = 5? / ¿Qué ecuación de valor absoluto corresponde?
|x − 2| = 5, since the center is 2 and the distance is 5 — Watch for: |x + 2| = 5, grabbed because a negative appears on the graph. Also |x − 5| = 2, from swapping center and distance. Center is the midpoint of the dots; distance is half the spread.
A = ½h(a + b). Find h when A = 45 cm², a = 6 cm, b = 12 cm. / Halla h cuando A = 45 cm², a = 6 cm, b = 12 cm.
h = 5 cm — Watch for: 2.5, from dividing 45 by 18 and forgetting the ½. Also 10, from doubling on the wrong side.
Evaluate (x² − y) ÷ (x + y) when x = 4 and y = −2. / Evalúa (x² − y) ÷ (x + y) cuando x = 4 y y = −2.
9 — Watch for: 5, from computing x² as 4 · 2. Also 7, from handling 16 − (−2) as 14.
Justify this step: 4(x + 9) becomes 4x + 36. (which property is this?)
/ Justifica el paso: 4(x + 9) se convierte en 4x + 36.
Distributive Property
Watch for: Associative, chosen because parentheses are involved. Associative regroups; distributive multiplies across.
A train leaves the station at 40 miles per hour. Three hours later a second train leaves on the same track at 60 miles per hour. Their distances from the station at time t are 40t and 60(t − 3). When are the trains the same distance from the station? / Las distancias desde la estación son 40t y 60(t − 3). ¿Cuándo están los trenes a la misma distancia?
t = 9 hours — Watch for: 3, from 40t = 60t − 3 with the 60 not distributed. Also 4.5, from subtracting 3 from t at the end instead of inside the parentheses. Also "they never will," picked when the algebra gets abandoned.
A stage is centered on the 50-yard line. Each outer edge is 14 yards from the center. Write an equation for the edge locations, then name both edges. / Escribe una ecuación para los bordes y nómbralos.
|x − 50| = 14; edges at 36 and 64 — Watch for: |x − 14| = 50, from swapping the center with the distance. Also 2|x − 50| = 14, from reading 14 as the full width of the stage.
Marcus used the conversion factors 1 ft/12 in and 60 min/1 hr. What could the starting and ending units of his rate be? / ¿Cuáles podrían ser las unidades inicial y final de su tasa?
Started in in/min, ended in ft/hr — Watch for: any choice ending in min/ft or hr/ft, where students flip the final rate. Push them to ask which unit cancels in each factor.
Classify 8 − v = v − 8 as an identity, one solution, or no solution. Justify with the solving steps. / Clasifica la ecuación y justifica con los pasos.
One solution, v = 8, since 8 − v = v − 8 gives 16 = 2v — Watch for: identity, chosen because both sides use the same two numbers. Also no solution, chosen because v sits on both sides. Only the solving steps settle it.