ACT study guide Math: Perform a Sanity Check - Avoid Errors

(9x−3)−(3x+6) is equivalent to

Error Buster: Remember to distribute the negative sign! (9x−3)−(3x+6) 9x−3−3x−6 6x−9 3(2x−3)

Josh rode his bicycle from his house to his grandmother's house at a rate of 10 miles per hour. Later that day he rode home at a rate of 15 miles per hour. If the round trip took a total of 30 minutes, how many miles does Josh live from his grandmother?

Part 1 You can write 2 individual distance = (rate)(time) equations, 1 for the Trip There and 1 for the Return Trip. Trip There d = (r)(t) d = Distance There r = 10 mph t = Time There Trip There equation: Distance There = (10 mph)(t) Return Trip d2 = (r2)(t2) d2 = Distance Back r2 = 15 mph t2 = Round Trip Time − Time There t2 = (.5 hours) − t Remember, t = Time There, from the Trip There Return Trip equation: Distance Back = (15 mph)(.5 − t) Part 2 We now have 2 d = (r)(t) equations: Trip There equation: Distance There = (10 mph)(t) Return Trip equation: Distance Back = (15 mph)(.5 − t) The key to Step 2 of this problem is that the distance is equal for both trips: Distance There = Distance Back Therefore, we can set the 2 d = (r)(t) equations equal to each other: (rate1)(time1)=(rate2)(time2) (15 mph)(.5 − t) = Distance Back = Distance There = (10 mph)(t) Eliminate the "Distance Back = Distance There", because they are no longer needed to get 1 equation with 1 variable, t: (10)(t)=(15)(0.5−t) 10t=7.5−15t 25t=7.5 t=0.3 hours Now remember, the question asks for how far Josh lives (d), NOT the time (t), so plug t=0.3 into d=rt and get d=(10 mph)(0.3 hours)=3 miles . Error Buster: Finish the question, don't stop with t = 0.3. Remember to ask, "Did I finish the question?"

If m=2p, then which of the following is equivalent to (m+4)^2

Plug 2p in for m and then FOIL. (2p+4)^2 (2p+4)(2p+4) 4p^2+8p+8p+16 4p^2+16p+16 Errors Buster: When you FOIL, remember the middle term!

If x=3y, then which of the following is equivalent to (x−2)2?

Plug in 3y for x in the expression (x−2)^2 to get ((3y)−2)^2=9y^2−12y+4 Error Buster: When you FOIL a binomial, remember the middle term.

When m=−1, which of the following is equivalent to m(2x^2−2)?

Plug in m=−1 and simplify. m(2x^2−2) (−1)(2x^2−2) −2x^2+2 Error Buster: Remember to distribute the negative sign!

If y=3a−15/x, find y when x=3

Plug x=3 into the equation to get y=3a−15/3=a−5, not a−15 . Error Buster: Remember to also divide the 15 by 3

If f(x)=x(2x^2−2), find f(−2p)

Plug −2p in for x . f(−2p)=(−2p)(2(−2p)^2−2) (−2p)(2(4p^2)−2) (−2p)(8p^2−2) −16p3+4p Error Buster: Remember to distribute the negative sign!

For functions p and r such that p(x)=2x(x−2) and r(x)=x3 find r(p(−4))

Plug −4 into the p function: p(−4)=2(−4)((−4)−2)=48 Then plug 48 into the r function: r(48)=48/3=16 Error Buster: Follow the order of operations.

If f(x)=3x2, which of the following expresses f(2p)

Remember the key to functions: whatever is in the parentheses gets plugged in for x. So, f(2p) just means plug in 2p for x : f(2p)=3(2p)2=3(4p^2)=12p^2 Error Busters: Remember that the 2 gets squared. Also remember order of operations, multiply 3 × 4 after squaring 2.

In a timed run, Maiko posted a time of 1 minute 45 seconds for a quarter-mile course. If she could maintain that pace, how many minutes would it take her to run 1 mile?

Since she ran a quarter (one-fourth) of a mile in 1 minute 45 seconds, to determine her time for a full mile, we multiply by 4. Error Buster: 1 minute 45 seconds times 4 is not (1.45)(4) = 5.8, it is (1.75)(4) = 7, since 45 seconds is three-quarters (0.75) of a minute.

If a board 9 feet 10 inches long is cut in half, how long is each new piece?

So easy to make a careless error, unless you know to watch for it. To divide 9 feet 10 inches by 2, do feet and inches separately. Error Buster: 9 feet divided by 2 is 4'6" , not 4'5", and 10 inches divided by 2 is 5". So 4'6" plus 5" equals 4'11"

If f(x)=3x2, which of the following expresses f(4p)?

f(4p) means plug 4p in for x. So f(4p)=3(4,p)^2=3(16p2)=48p^2 . Error Buster: Remember that the 4 gets squared before you multiply by 3. PEMDAS.

If f(x)=−2x3, which of the following expresses f(−p)?

f(−p) means plug −p in for x . f(−p)=−2(−p)^3 −2(−p^3) 2p^3 Error Buster: Remember that the 2 is not in the parenthesis and does not get cubed with the p .