Inequalities: Phone Plans

Which phone plan to choose?  A little math tutoring can help you decide.

Let’s assume we have two choices:

Plan A $12 per month $0.10 per text
Plan B $18 per month $0.07 per text

How do you decide which plan to use?

Let’s say you’re buying the plan for a month.  Let x be the number of texts you send.  Then your cost is

$12 + $0.10x for Plan A

$18 + $0.07x for Plan B

You pay more up front with Plan B, but its texts are cheaper.  There will be a specific number of texts at which the plans cost the same:

12 + 0.10x =18 + 0.07x

Using algebra, we solve for x:

  1. Subtract 0.07x from both sides to get

    12 + 0.03x = 18

  2. Subtract 12 from from both sides:


  3. Divide both sides by 0.03:


At 200 texts, the two plans are equal in price for one month.  Any more texts, and Plan B must be cheaper:  it’s less per text.

Now, let’s explore the situation from the point of view of inequalities.  We ask, “For what number of texts will Plan B be cheaper?”

18 + 0.07x ≤ 12 + 0.10x

  1. Subtract 18 from both sides

    0.07x ≤ -6 + 0.10x

  2. Subtract 0.10x from both sides

    -0.03x ≤ -6

  3. Divide both sides by -0.03: you must flip the sign when doing so:

    x ≥ 200

Note that algebraic operations are the same for inequalities as for equations, with one important exception:  you must flip the sign if you divide or multiply by a negative number.  (Not by positive, just by negative).

The two methods make the same conclusion, but from different points of view.  The equation tells us that, at 200 texts, the two plans cost the same.  The inequality, on the other hand, tells us that if we make more than 200 texts, we’ll pay less with Plan B.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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