12 multiple-choice questions, progressively harder.
Simplify: 3(x+2)−(x−4)3(x + 2) - (x - 4)3(x+2)−(x−4).
Solution
Correct answer: A
Distribute the 333, and treat the leading minus as a factor of −1-1−1 that flips both terms inside the second group.
3x+6−x+4=2x+103x + 6 - x + 4 = 2x + 103x+6−x+4=2x+10
The −4-4−4 becomes +4+4+4 when the sign flips, so the constants give 6+4=106 + 4 = 106+4=10.
Evaluate 3a−2b3a - 2b3a−2b when a=−4a = -4a=−4 and b=−5b = -5b=−5.
Correct answer: C
Substitute both values in parentheses, do each multiplication with the sign rules, then add.
3(−4)−2(−5)=−12+10=−23(-4) - 2(-5) = -12 + 10 = -23(−4)−2(−5)=−12+10=−2
The term 3(−4)=−123(-4) = -123(−4)=−12, and −2(−5)=+10-2(-5) = +10−2(−5)=+10, so the total is −2-2−2.
Evaluate 2x2+52x^2 + 52x2+5 when x=−4x = -4x=−4.
Correct answer: B
Substitute −4-4−4 in parentheses, square first, then multiply, then add.
2(−4)2+5=2(16)+5=372(-4)^2 + 5 = 2(16) + 5 = 372(−4)2+5=2(16)+5=37
The parentheses make (−4)2=16(-4)^2 = 16(−4)2=16 positive, so 2×16=322 \times 16 = 322×16=32 and 32+5=3732 + 5 = 3732+5=37.
Evaluate (x+y)2(x + y)^2(x+y)2 when x=3x = 3x=3 and y=2y = 2y=2.
Correct answer: D
Substitute both values, finish inside the parentheses first, then square the result.
((3)+(2))2=(5)2=25\big((3) + (2)\big)^2 = (5)^2 = 25((3)+(2))2=(5)2=25
The grouping forces the addition 3+2=53 + 2 = 53+2=5 before the squaring; this is not 32+22=133^2 + 2^2 = 1332+22=13.
Simplify: x2+3x+2x2−xx^2 + 3x + 2x^2 - xx2+3x+2x2−x.
Combine the x2x^2x2-terms with each other and the xxx-terms with each other; the two are unlike.
(x2+2x2)+(3x−x)=3x2+2x(x^2 + 2x^2) + (3x - x) = 3x^2 + 2x(x2+2x2)+(3x−x)=3x2+2x
The x2x^2x2-terms give 3x23x^23x2 and the xxx-terms give 2x2x2x; powers are never added when combining.
Simplify: 13(9x−12)+2x\dfrac{1}{3}(9x - 12) + 2x31(9x−12)+2x.
Distribute the 13\tfrac{1}{3}31 across both terms, then combine the xxx-terms.
3x−4+2x=5x−43x - 4 + 2x = 5x - 43x−4+2x=5x−4
A third of 9x9x9x is 3x3x3x and a third of −12-12−12 is −4-4−4; then 3x+2x=5x3x + 2x = 5x3x+2x=5x.
Simplify: 5(x−2)−4(x−3)5(x - 2) - 4(x - 3)5(x−2)−4(x−3).
Distribute each factor, watching that the −4-4−4 flips the signs of the second group.
(5x−10)+(−4x+12)=x+2(5x - 10) + (-4x + 12) = x + 2(5x−10)+(−4x+12)=x+2
The −4-4−4 times −3-3−3 gives +12+12+12, so the constants total −10+12=2-10 + 12 = 2−10+12=2.
Evaluate x2−1x+1\dfrac{x^2 - 1}{x + 1}x+1x2−1 when x=4x = 4x=4.
Substitute 444 for each xxx, finishing the numerator and denominator before dividing.
(4)2−1(4)+1=155=3\frac{(4)^2 - 1}{(4) + 1} = \frac{15}{5} = 3(4)+1(4)2−1=515=3
The fraction bar groups top and bottom, so 16−1=1516 - 1 = 1516−1=15 over 4+1=54 + 1 = 54+1=5 gives 333. Forgetting the −1-1−1 would wrongly give 165\tfrac{16}{5}516.
Simplify: 4x+3(x+2)−5x4x + 3(x + 2) - 5x4x+3(x+2)−5x.
Distribute the 333, then combine all three xxx-terms and keep the constant.
4x+3x+6−5x=2x+64x + 3x + 6 - 5x = 2x + 64x+3x+6−5x=2x+6
The xxx-terms give 4x+3x−5x=2x4x + 3x - 5x = 2x4x+3x−5x=2x and the constant 666 stays.
Simplify: 5(x−1)−2(2x−3)+45(x - 1) - 2(2x - 3) + 45(x−1)−2(2x−3)+4.
Distribute each factor, watching the −2-2−2, then combine the xxx-terms and the constants.
5x−5−4x+6+4=x+55x - 5 - 4x + 6 + 4 = x + 55x−5−4x+6+4=x+5
The xxx-terms give 5x−4x=x5x - 4x = x5x−4x=x and the constants give −5+6+4=5-5 + 6 + 4 = 5−5+6+4=5.
Simplify: 12(8x−6)+5\dfrac{1}{2}(8x - 6) + 521(8x−6)+5.
Distribute the 12\tfrac{1}{2}21 across both terms, then add the constant.
4x−3+5=4x+24x - 3 + 5 = 4x + 24x−3+5=4x+2
Half of 8x8x8x is 4x4x4x and half of −6-6−6 is −3-3−3, then −3+5=2-3 + 5 = 2−3+5=2.
Simplify: 7−2(x+3)−(x−5)7 - 2(x + 3) - (x - 5)7−2(x+3)−(x−5).
Distribute the −2-2−2, flip both terms of the last group, then combine like terms.
7−2x−6−x+5=−3x+67 - 2x - 6 - x + 5 = -3x + 67−2x−6−x+5=−3x+6
The xxx-terms give −2x−x=−3x-2x - x = -3x−2x−x=−3x and the constants give 7−6+5=67 - 6 + 5 = 67−6+5=6.
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