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Multi-Step Inequalities

Key Questions

  • I would start with arranging terms so that all variables are on one side.


    I hope that this was helpful.

  • Answer:

    #There are generally 3 methods to solve inequalities

    Explanation:

    We can usually solve inequalities by 3 ways:

    1. By algebraic method
      Example 1: Solve: 2x - 7 < x - 5
      2x - x < 7 - 5
      x < 2
    2. By the number-line method.
      Example 2. Solve #f(x) = x^2 + 2x - 3 < 0#
      First, solve f(x) = 0. There are 2 real roots x1 = 1, and x2 = - 3.
      Replace x = 0 into f(x). We find f(0) = - 3 < 0. Therefor, the origin O is located inside the solution set.
      Answer by interval: (- 3, 1)

    --------------------- - 3 ++++++++ 0 ++++ 1 ----------------

    3.By graphing method.
    Example 2. Solve: #f(x) = x^2 + 2x - 3 < 0#.
    The graph of f(x) is an upward parabola (a > 0), that intersects the x-axis at x1 = 1 and x2 = - 3. Inside the interval (-3, 1), the parabola stays below the x-axis --> f(x) < 0.
    Therefor, the solution set is the open interval (-3, 1)
    graph{x^2 + 2x - 3 [-10, 10, -5, 5]}

  • Suppose we're solving in #\mathbb{R}#

    #0x=1 \Rightarrow S = \emptyset#

    #0x=0 \Rightarrow S = \mathbb{R}#

  • Answer:

    Inequalities are very tricky.

    Explanation:

    When solving a multi step equation, you use PEMDAS (parentheses, exponents, multiplication, division, add, subtract), and you also use PEMDAS when solving a multi step inequality. However, inequalities are tricky in the fact that if you multiply or divide by a negative number, you must flip the sign. And while normally there are 1 or 2 solutions to a multi step equation, in the form of x= #, you'll have the same thing, but with an inequality sign (or signs).

Questions