Maths-
General
Easy

Question

Let a, b, c and x be real numbers. How is solving vertical line a x vertical line plus b less or equal than c different from solving vertical line a x plus b vertical line less or equal than c

The correct answer is: -(b+c)/a≤x and x≤-(b-c)/a Finally, we can observe that the solutions we get from the two given inequalities are different.


    Hint:
    |x| is known as the absolute value of x. It is the non-negative value of x irrespective of its sign. The value of absolute value of x is given by
    vertical line x vertical line equals open curly brackets table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell negative x comma x less than 0 end cell row cell x comma x greater or equal than 0 end cell end table close
    The symbol |.| is pronounced as ‘modulus’. We read |x| as ‘modulus of x’ or ‘mod x’.
    We discuss both these cases for the two given inequalities.
    Step by step solution:
    We assume that a, b, c are positive.
    The first inequality is
    |ax| + b ≤ c
    Applying the definition of the modulus for ax in the above inequality, we get
    For ax < 0, we have
    -ax + b ≤ c
    ⇒ -ax ≤ c - b
    ⇒ ax ≥ b - c
    Then solving for x, we get
    x greater or equal than fraction numerator b minus c over denominator a end fraction
    For ax ≥ 0, we have
    ax + b ≤ c
    Solving for x, we get
    x less or equal than fraction numerator c minus b over denominator a end fraction
    Or
    x less or equal than negative fraction numerator b minus c over denominator a end fraction
    Combining the two solutions, we have
    x greater or equal than fraction numerator b minus c over denominator a end fraction text  and  end text x less or equal than negative fraction numerator b minus c over denominator a end fraction
    The second inequality is
    |ax + b| ≤ c
    We use the definition of modulus,
    For ax + b < 0, we have
    -(ax + b) ≤ c
    ⇒ -ax - b ≤ c
    ⇒ -ax ≤ b + c
    Solving for x, we get
    x greater or equal than negative fraction numerator b plus c over denominator a end fraction
    For ax + b ≥ 0, we have
    ax + b ≤ c
    ⇒ax ≤ c - b
    Solving for x, we get
    x less or equal than negative fraction numerator b minus c over denominator a end fraction
    Combining the two solutions we get
    negative fraction numerator b plus c over denominator a end fraction less or equal than x text  and  end text x less or equal than negative fraction numerator b minus c over denominator a end fraction
    Finally, we can observe that the solutions we get from the two given inequalities are different.
    Note:
    In the first inequalities, mod changes the value of only one term, i.e., ax; whereas in the second inequality, mod changes the value two terms, ax and b. This results in the change in values of x for both inequalities.

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