24-(DECEMBER $$ 2005 / 0 $$ 48) MEDIUM Which of the following gives the range of numbers that are within 1.5 of the number $$ \frac{2}{3} $$ ? E. $$ -\frac{13}{6} $$ to $$ \frac{5}{6} $$ G. $$ -\frac{13}{6} $$ to $$ \frac{13}{6} $$ H. $$ -\frac{5}{6} $$ to $$ \cdot \frac{5}{6} $$ J. $$ -\frac{5}{6} $$ to $$ \frac{13}{6} $$ K. -1 to 1

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To determine the range of numbers that are within $$1.5$$ of $$\frac{2}{3}$$, we set up the inequality

$$
\left|x-\frac{2}{3}\right|\leq 1.5.
$$

This inequality means that $$x$$ can be at most $$1.5$$ away from $$\frac{2}{3}$$.

1. Write the inequality without absolute value:

$$
-\;1.5 \leq x-\frac{2}{3} \leq 1.5.
$$

2. Next, add $$\frac{2}{3}$$ to each part of the inequality to solve for $$x$$:

$$
\frac{2}{3}-1.5 \leq x \leq \frac{2}{3}+1.5.
$$

3. Convert $$1.5$$ to fraction form for ease of computation: $$1.5=\frac{3}{2}$$.

4. Express $$\frac{2}{3}$$ and $$\frac{3}{2}$$ with a common denominator. The common denominator for $$3$$ and $$2$$ is $$6$$:

$$
\frac{2}{3}=\frac{4}{6} \quad \text{and} \quad \frac{3}{2}=\frac{9}{6}.
$$

5. Now compute the lower bound:

$$
\frac{4}{6} - \frac{9}{6} = -\frac{5}{6}.
$$

6. Compute the upper bound:

$$
\frac{4}{6} + \frac{9}{6} = \frac{13}{6}.
$$

Thus, the range of numbers is from $$-\frac{5}{6}$$ to $$\frac{13}{6}$$.

Comparing with the provided options, the correct one is:

J. $$-\frac{5}{6}$$ to $$\frac{13}{6}$$.
The range of numbers within 1.5 of $$\frac{2}{3}$$ is from $$-\frac{5}{6}$$ to $$\frac{13}{6}$$.

24-(DECEMBER 48) MEDIUM
Which of the following gives the range of numbers that
are within 1.5 of the number ?
E. to
G. to
H. to
J. to
K. -1 to 1

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24-(DECEMBER 48) MEDIUM Which of the following gives the range of numbers that are within 1.5 of the number ? E. to G. to H. to J. to K. -1 to 1