12 multiple-choice questions, progressively harder.
Solve x4=2.5\dfrac{x}{4} = 2.54x=2.5 for xxx.
Solution
Correct answer: A
Multiply both sides by 444 to undo the division.
4⋅x4=4⋅2.5 ⇒ x=104 \cdot \frac{x}{4} = 4 \cdot 2.5 \;\Rightarrow\; x = 104⋅4x=4⋅2.5⇒x=10
Check: 104=2.5\frac{10}{4} = 2.5410=2.5, which is true.
Solve 56x=20\dfrac{5}{6}x = 2065x=20 for xxx.
Correct answer: C
Multiply both sides by the reciprocal 65\tfrac{6}{5}56 to undo multiplication by 56\tfrac{5}{6}65.
65⋅56x=65⋅20 ⇒ x=24\frac{6}{5} \cdot \frac{5}{6}x = \frac{6}{5} \cdot 20 \;\Rightarrow\; x = 2456⋅65x=56⋅20⇒x=24
Check: 56(24)=1206=20\tfrac{5}{6}(24) = \tfrac{120}{6} = 2065(24)=6120=20, which is true.
Solve 35x=9\dfrac{3}{5}x = 953x=9 for xxx.
Correct answer: D
Multiply both sides by the reciprocal 53\tfrac{5}{3}35 to undo multiplication by 35\tfrac{3}{5}53.
53⋅35x=53⋅9 ⇒ x=15\frac{5}{3} \cdot \frac{3}{5}x = \frac{5}{3} \cdot 9 \;\Rightarrow\; x = 1535⋅53x=35⋅9⇒x=15
Check: 35(15)=455=9\tfrac{3}{5}(15) = \tfrac{45}{5} = 953(15)=545=9, which is true.
Solve x+16=56x + \tfrac{1}{6} = \tfrac{5}{6}x+61=65 for xxx.
Correct answer: B
Subtract 16\tfrac{1}{6}61 from both sides; the fractions share the denominator 666.
x=56−16=46=23x = \tfrac{5}{6} - \tfrac{1}{6} = \tfrac{4}{6} = \tfrac{2}{3}x=65−61=64=32
Check: 23+16=46+16=56\tfrac{2}{3} + \tfrac{1}{6} = \tfrac{4}{6} + \tfrac{1}{6} = \tfrac{5}{6}32+61=64+61=65, which is true.
Solve x−3=5\dfrac{x}{-3} = 5−3x=5 for xxx.
The variable is divided by −3-3−3, so multiply both sides by −3-3−3.
−3⋅x−3=−3⋅5 ⇒ x=−15-3 \cdot \frac{x}{-3} = -3 \cdot 5 \;\Rightarrow\; x = -15−3⋅−3x=−3⋅5⇒x=−15
Check: −15−3=5\frac{-15}{-3} = 5−3−15=5, which is true.
Solve x−9=−9x - 9 = -9x−9=−9 for xxx.
Add 999 to both sides.
x−9+9=−9+9 ⇒ x=0x - 9 + 9 = -9 + 9 \;\Rightarrow\; x = 0x−9+9=−9+9⇒x=0
Check: 0−9=−90 - 9 = -90−9=−9, which is true.
Solve x−23=−13x - \tfrac{2}{3} = -\tfrac{1}{3}x−32=−31 for xxx.
Add 23\tfrac{2}{3}32 to both sides; the fractions share the denominator 333.
x=−13+23=13x = -\tfrac{1}{3} + \tfrac{2}{3} = \tfrac{1}{3}x=−31+32=31
Check: 13−23=−13\tfrac{1}{3} - \tfrac{2}{3} = -\tfrac{1}{3}31−32=−31, which is true.
Solve −34x=9-\tfrac{3}{4}x = 9−43x=9 for xxx.
Multiply both sides by the reciprocal of −34-\tfrac{3}{4}−43, which is −43-\tfrac{4}{3}−34.
−43⋅(−34x)=−43⋅9 ⇒ x=−12-\frac{4}{3} \cdot \left(-\frac{3}{4}x\right) = -\frac{4}{3} \cdot 9 \;\Rightarrow\; x = -12−34⋅(−43x)=−34⋅9⇒x=−12
Check: −34(−12)=364=9-\tfrac{3}{4}(-12) = \tfrac{36}{4} = 9−43(−12)=436=9, which is true.
For which equation does multiplying both sides by 555 isolate the variable?
Multiplying by 555 undoes a division by 555, so the variable must be divided by 555.
5⋅x5=5⋅30 ⇒ x=1505 \cdot \frac{x}{5} = 5 \cdot 30 \;\Rightarrow\; x = 1505⋅5x=5⋅30⇒x=150
The other forms call for subtracting 555, dividing by 555, or adding 555, not multiplying.
Solve 30=6x30 = 6x30=6x for xxx.
Divide both sides by 666, even though the variable term is on the right.
306=6x6 ⇒ 5=x\frac{30}{6} = \frac{6x}{6} \;\Rightarrow\; 5 = x630=66x⇒5=x
Check: 6(5)=306(5) = 306(5)=30, which is true.
Solve 11x=−4411x = -4411x=−44 for xxx.
Divide both sides by 111111.
11x11=−4411 ⇒ x=−4\frac{11x}{11} = \frac{-44}{11} \;\Rightarrow\; x = -41111x=11−44⇒x=−4
Check: 11(−4)=−4411(-4) = -4411(−4)=−44, which is true.
Solve −1.5=x+2.5-1.5 = x + 2.5−1.5=x+2.5 for xxx.
Subtract 2.52.52.5 from both sides, even though the variable is on the right.
−1.5−2.5=x+2.5−2.5 ⇒ −4=x-1.5 - 2.5 = x + 2.5 - 2.5 \;\Rightarrow\; -4 = x−1.5−2.5=x+2.5−2.5⇒−4=x
Check: −4+2.5=−1.5-4 + 2.5 = -1.5−4+2.5=−1.5, which is true.
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