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jmc

algebra intermediate

Problem

Express, in simplest form, the value of

\frac{3}{4} - \frac{1}{2} - \frac{3}{4} + \frac{1}{2} .

Solution

Let

x = \frac{3}{4} - \frac{1}{2} - \frac{3}{4} + \frac{1}{2} .

Then

x^{2} = \frac{3}{4} - \frac{1}{2} - 2 \frac{3}{4} - \frac{1}{2} \frac{3}{4} + \frac{1}{2} + \frac{3}{4} + \frac{1}{2} = \frac{3}{2} - 2 \frac{9}{16} - \frac{1}{2} = \frac{3}{2} - 2 \frac{1}{16} = \frac{3}{2} - \frac{1}{2} = 1.

Since

\frac{3}{4} + \frac{1}{2} > \frac{3}{4} - \frac{1}{2},

x

is negative, so

x = - 1 .

Final answer

-1