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Difficulty: MediumLinear Equations in One Variable

If xx satisfies the equation 3x−12−x+45=x+710+2\frac{3x - 1}{2} - \frac{x + 4}{5} = \frac{x + 7}{10} + 2, which of the following statements about xx must be true? Indicate all such statements.

  1. x>3x > 3Answer
  2. 3x3x is an integerAnswer
  3. xx is a solution to the equation 3x−10=03x - 10 = 0Answer
  4. D
    xx is an integer
  5. E
    x<3.2x < 3.2

Answer

The statements that x>3x > 3, that 3x3x is an integer, and that xx is a solution to the equation 3x−10=03x - 10 = 0 are all correct.
Solving the equation yields x=103≈3.33x = \frac{10}{3} \approx 3.33. The statement x>3x > 3 is correct since 3.33>33.33 > 3. The statement that 3x3x is an integer is correct because 3×103=103 \times \frac{10}{3} = 10. The statement that xx is a solution to 3x−10=03x - 10 = 0 is correct because 3(103)−10=03\left(\frac{10}{3}\right) - 10 = 0.

Step-by-Step Solution

1
Clear denominators by multiplying the entire linear equation by the least common multiple, 10.
5(3x−1)−2(x+4)=(x+7)+205(3x - 1) - 2(x + 4) = (x + 7) + 20
Eliminating fractions simplifies the expression into a standard linear form.
2
Distribute terms across parentheses.
15x−5−2x−8=x+7+2015x - 5 - 2x - 8 = x + 7 + 20
Expand both sides while correctly applying sign rules.
3
Combine like terms on each side.
13x−13=x+2713x - 13 = x + 27
Group variable terms together and constant terms together.
4
Isolate the variable xx.
12x=40  ⟹  x=4012=10312x = 40 \implies x = \frac{40}{12} = \frac{10}{3}
Subtract xx and add 1313 on both sides, then divide by 1212.
5
Evaluate each given statement using x=103x = \frac{10}{3}.
The value 103≈3.33\frac{10}{3} \approx 3.33 is greater than 33; 3(103)=103\left(\frac{10}{3}\right) = 10 is an integer; 3(103)−10=03\left(\frac{10}{3}\right) - 10 = 0 is true; 103\frac{10}{3} is not an integer; 103\frac{10}{3} is not less than 3.23.2.
Determine which logical assertions hold true for x=103x = \frac{10}{3}.

Key Concept

Solving multi-step linear equations with fractions and evaluating properties of rational solutions.
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Topics

Linear Equations in One Variable

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