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The lengths of three sides of a triangle are given by (x + 7), (9 - 2x) and (3x - 1). For how many integral values of x will the triangle be isosceles? 
    Correct answer is '2'. Can you explain this answer?
    Verified Answer
    The lengths of three sides of a triangle are given by (x + 7), (9 - 2x...
    Solution: An isosceles triangle has two sides equal. We have three cases:
    Case (i): x + 7 = 9 - 2x
    3x = 2. x = 2/3, which is not an integer.
    Case (ii): x + 7 =  3x - 1
    2x = 8,  x = 4
    The sides are 11,11 and 1
    Case (iii): 9 - 2x = 3x - 1, 5x = 10, x = 2 The sides are 5, 5 and 9
    The triangle is isosceles for two values of x. 
    Answer: 2
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    Most Upvoted Answer
    The lengths of three sides of a triangle are given by (x + 7), (9 - 2x...
    Given:
    The lengths of three sides of a triangle are given by (x + 7), (9 - 2x), and (3x - 1).

    To find:
    The number of integral values of x for which the triangle is isosceles.

    Explanation:
    A triangle is said to be isosceles if at least two sides of the triangle are equal in length.

    In this case, we need to find the values of x for which two sides of the triangle are equal.

    Step 1:
    Set up the equation to represent the condition for an isosceles triangle.

    (x + 7) = (9 - 2x) or (9 - 2x) = (3x - 1) or (3x - 1) = (x + 7)

    Step 2:
    Solve each equation separately to find the values of x.

    For the first equation,
    x + 7 = 9 - 2x
    3x = 2
    x = 2/3

    For the second equation,
    9 - 2x = 3x - 1
    5x = 10
    x = 2

    For the third equation,
    3x - 1 = x + 7
    2x = 8
    x = 4

    Step 3:
    Check if the values of x satisfy the conditions for an isosceles triangle.

    For x = 2/3, the lengths of the sides are (x + 7) = 19/3, (9 - 2x) = 17/3, and (3x - 1) = 5/3. Since none of the sides are equal, this value of x does not satisfy the condition for an isosceles triangle.

    For x = 2, the lengths of the sides are (x + 7) = 9, (9 - 2x) = 5, and (3x - 1) = 5. The triangle formed by these side lengths is isosceles.

    For x = 4, the lengths of the sides are (x + 7) = 11, (9 - 2x) = 1, and (3x - 1) = 11. The triangle formed by these side lengths is isosceles.

    Conclusion:
    Out of the three values of x, only two values (x = 2 and x = 4) satisfy the condition for an isosceles triangle. Therefore, the number of integral values of x for which the triangle is isosceles is 2.
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    The lengths of three sides of a triangle are given by (x + 7), (9 - 2x) and (3x - 1). For how many integral values of x will the triangle be isosceles?Correct answer is '2'. Can you explain this answer?
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