12 multiple-choice questions, progressively harder.
How many solutions does 2(3x+1)=6x−52(3x + 1) = 6x - 52(3x+1)=6x−5 have?
Solution
Correct answer: D
Distribute the left side.
6x+2=6x−56x + 2 = 6x - 56x+2=6x−5
Subtracting 6x6x6x leaves 2=−52 = -52=−5, which is false, so the equation has no solution.
Solve 2x−35=3\dfrac{2x - 3}{5} = 352x−3=3.
Multiply both sides by 555 to clear the denominator.
2x−3=15,2x=18,x=92x - 3 = 15, \qquad 2x = 18, \qquad x = 92x−3=15,2x=18,x=9
Solve the proportion x+22=2x−13\dfrac{x + 2}{2} = \dfrac{2x - 1}{3}2x+2=32x−1.
Correct answer: C
Cross-multiply.
3(x+2)=2(2x−1),3x+6=4x−2,8=x3(x + 2) = 2(2x - 1), \qquad 3x + 6 = 4x - 2, \qquad 8 = x3(x+2)=2(2x−1),3x+6=4x−2,8=x
How many solutions does 2(4x−3)=8x−62(4x - 3) = 8x - 62(4x−3)=8x−6 have?
Correct answer: B
8x−6=8x−68x - 6 = 8x - 68x−6=8x−6
Subtracting 8x8x8x leaves −6=−6-6 = -6−6=−6, true for every xxx, so there are infinitely many solutions.
Solve 3(x−2)−2(x−1)=53(x - 2) - 2(x - 1) = 53(x−2)−2(x−1)=5.
Distribute both products and combine like terms.
3x−6−2x+2=5,x−4=5,x=93x - 6 - 2x + 2 = 5, \qquad x - 4 = 5, \qquad x = 93x−6−2x+2=5,x−4=5,x=9
How many solutions does 5(x+2)=5x+75(x + 2) = 5x + 75(x+2)=5x+7 have?
Correct answer: A
5x+10=5x+75x + 10 = 5x + 75x+10=5x+7
Subtracting 5x5x5x leaves 10=710 = 710=7, which is false, so the equation has no solution.
Solve the proportion 2x+13=x+42\dfrac{2x + 1}{3} = \dfrac{x + 4}{2}32x+1=2x+4.
2(2x+1)=3(x+4),4x+2=3x+12,x=102(2x + 1) = 3(x + 4), \qquad 4x + 2 = 3x + 12, \qquad x = 102(2x+1)=3(x+4),4x+2=3x+12,x=10
Solve x−2x+2=3\dfrac{x - 2}{x + 2} = 3x+2x−2=3, given x≠−2x \neq -2x=−2.
x−2=3(x+2),x−2=3x+6,−8=2x,x=−4x - 2 = 3(x + 2), \qquad x - 2 = 3x + 6, \qquad -8 = 2x, \qquad x = -4x−2=3(x+2),x−2=3x+6,−8=2x,x=−4
Since −4≠−2-4 \neq -2−4=−2, the solution is x=−4x = -4x=−4.
Solve x4−x−26=2\dfrac{x}{4} - \dfrac{x - 2}{6} = 24x−6x−2=2.
The LCD of 444 and 666 is 121212. Multiply every term by 121212, keeping x−2x - 2x−2 grouped.
3x−2(x−2)=24,3x−2x+4=24,x+4=24,x=203x - 2(x - 2) = 24, \qquad 3x - 2x + 4 = 24, \qquad x + 4 = 24, \qquad x = 203x−2(x−2)=24,3x−2x+4=24,x+4=24,x=20
Solve 4x−1=2\dfrac{4}{x - 1} = 2x−14=2, given x≠1x \neq 1x=1.
Multiply both sides by x−1x - 1x−1 to clear the denominator.
4=2(x−1),4=2x−2,2x=6,x=34 = 2(x - 1), \qquad 4 = 2x - 2, \qquad 2x = 6, \qquad x = 34=2(x−1),4=2x−2,2x=6,x=3
Since 3≠13 \neq 13=1, it is a valid solution.
How many solutions does 3(x−1)−x=2x−33(x - 1) - x = 2x - 33(x−1)−x=2x−3 have?
Simplify the left side.
3x−3−x=2x−3,2x−3=2x−33x - 3 - x = 2x - 3, \qquad 2x - 3 = 2x - 33x−3−x=2x−3,2x−3=2x−3
Subtracting 2x2x2x leaves −3=−3-3 = -3−3=−3, true for every xxx, so there are infinitely many solutions.
Solve the proportion x−26=x+19\dfrac{x - 2}{6} = \dfrac{x + 1}{9}6x−2=9x+1.
9(x−2)=6(x+1),9x−18=6x+6,3x=24,x=89(x - 2) = 6(x + 1), \qquad 9x - 18 = 6x + 6, \qquad 3x = 24, \qquad x = 89(x−2)=6(x+1),9x−18=6x+6,3x=24,x=8
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