[(3x-6)/(4x-8)]<=1. Solve; use interval notation?

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  • 1 month ago

    (3x - 6)/(4x - 8) ≤ 1

    [3(x - 2)]/[4(x - 2)] ≤ 1

    3/4 ≤ 1  and  x - 2 ≠ 0   (because 0/0 is undefined)

    For all values of x, 3/4 < 1

    Hence, -∞ < x < ∞  and  x ≠ 2

    i.e. -∞ < x < 2 or 2 < x < ∞

    The answer in interval notation: x ∊ (-∞, 2) or (2, ∞)

  • sepia
    Lv 7
    1 month ago

     [(3x - 6)/(4x - 8)] ≤ 1

    Solutions:

    2 > x > 2

  • atsuo
    Lv 6
    1 month ago

    If x = 2 then (3x-6)/(4x-8) = 0/0 is indefinite.

    So x = 2 is not a solution.

    If x < 2 then 4x-8 < 0, so

    (3x-6)/(4x-8) ≦ 1

    3x-6 ≧ 4x-8

    2 ≧ x

    But x < 2 must be satisfied, so the solution is x < 2.

    If x > 2 then 4x-8 > 0, so

    (3x-6)/(4x-8) ≦ 1

    3x-6 ≦ 4x-8

    2 ≦ x

    But x > 2 must be satisfied, so the solution is x > 2.

    Therefore, the solution is x < 2 or x > 2.

    Using interval notation, (-inf,2) ∪ (2,inf).

  • 1 month ago

    [(3x - 6)/(4x - 8)] <= 1

    3x - 6 <= 4x - 8

    x >= 2

    2 < x > 2

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  • 1 month ago

    [(3x - 6)/(4x - 8)] ≤ 1

    [(3x - 6)/(4x - 8)] - 1 ≤ 0

    [(3x - 6) - (4x - 8)]/(4x - 8) ≤ 0

    (3x - 6 - 4x + 8)/(4x - 8) ≤ 0

    (- x + 2)/(4x - 8) ≤ 0

    (- x + 2)/[4.(x - 2)] ≤ 0 → you know that: 4 > 0

    (- x + 2)/(x - 2) ≤ 0

    - (x - 2)/(x - 2) ≤ 0

    - 1 ≤ 0 ← wrong → no solution

  • ?
    Lv 7
    1 month ago

    3x-6

    ------ ≤ 1 ⇒

    4x-8

    // Multiply both sides by (4x-8)

    // to get rid of fraction

    3x-6 ≤ 4x-8

    // Combine like-terms: Subtract

    // 3x from both sides

    3x-3x - 6 ≤ 4x-3x - 8

    -6 ≤ x - 8

    // Add 8 to both sides

    -6 + 8 ≤ x - 8 + 8

    2 ≤ x

    Solution in interval notation is [2,+∞).........ANS

  • TomV
    Lv 7
    1 month ago

    [(3x-6)/(4x-8)] ≤ 1

    [3(x-2)]/[4(x-2)] ≤ 1

    (3/4) ≤ 1

    True for all values of x.

    The value of the rational expression has to be considered in the limit as x→2 since the expression evaluates to the indeterminate value 0/0. But, since that limit exists and is equal to 3/4, the solution set also includes x = 2

  • 1 month ago

    (3x - 6)/(4x - 8) ≤ 1

    We need to multiply both sides by (4x - 8)² to make sure we have a positive value and the inequality stays the same.

    so, (3x - 6)(4x - 8) ≤ (4x - 8)²

    i.e. (4x - 8)² - (3x - 6)(4x - 8) ≥ 0 

    so, (4x - 8)[(4x - 8) - (3x - 6)]  ≥ 0

    so, (4x - 8)(x - 2) ≥ 0

    i.e. (x - 2)(x - 2) ≥ 0

    or, (x - 2)² ≥ 0....true for all values of x

    As we cannot have x = 2, the solution would be (-∞, 2) ∪ (2, ∞)

    :)>  

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