To find the distance between the two parallel lines [tex]\(\ell_1\)[/tex] and [tex]\(\ell_2\)[/tex] given by the equations [tex]\(y = -x - 3\)[/tex] and [tex]\(y = -x + \frac{2}{5}\)[/tex], we use the formula for the distance between two parallel lines [tex]\(y = mx + c_1\)[/tex] and [tex]\(y = mx + c_2\)[/tex]:
[tex]\[
\text{distance} = \frac{|c_1 - c_2|}{\sqrt{1 + m^2}}
\][/tex]
Here, both lines have the same slope [tex]\(m = -1\)[/tex].
1. Identify [tex]\(c_1\)[/tex] and [tex]\(c_2\)[/tex] from the equations:
- For [tex]\(\ell_1\)[/tex], the line equation is [tex]\(y = -x - 3\)[/tex], so [tex]\(c_1 = -3\)[/tex].
- For [tex]\(\ell_2\)[/tex], the line equation is [tex]\(y = -x + \frac{2}{5}\)[/tex], so [tex]\(c_2 = \frac{2}{5}\)[/tex].
2. Calculate the difference [tex]\(|c_1 - c_2|\)[/tex]:
[tex]\[
|c_1 - c_2| = \left| -3 - \frac{2}{5} \right| = \left| -3 - 0.4 \right| = \left| -3.4 \right| = 3.4
\][/tex]
3. Calculate the denominator [tex]\(\sqrt{1 + m^2}\)[/tex]:
[tex]\[
\sqrt{1 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\][/tex]
4. Substitute these values into the distance formula:
[tex]\[
\text{distance} = \frac{3.4}{\sqrt{2}}
\][/tex]
5. Compute the division:
[tex]\[
\frac{3.4}{\sqrt{2}} \approx 2.4041630560342613
\][/tex]
6. Round the result to the nearest tenth:
[tex]\[
2.4041630560342613 \approx 2.4
\][/tex]
Therefore, the distance between the lines [tex]\(\ell_1\)[/tex] and [tex]\(\ell_2\)[/tex] is approximately [tex]\(2.4\)[/tex] when rounded to the nearest tenth.
[tex]\[
\boxed{2.4}
\][/tex]
