To evaluate the expression [tex]\(\frac{4}{15} + x + 0.4\)[/tex] for different values of [tex]\(x\)[/tex], let's go through each case step by step.
Case 1: [tex]\( x = -1 \)[/tex]
1. Substitute [tex]\( x = -1 \)[/tex] into the expression:
[tex]\[
\frac{4}{15} + (-1) + 0.4
\][/tex]
2. Simplify the expression step by step:
[tex]\[
\frac{4}{15} - 1 + 0.4
\][/tex]
3. Combine the constants:
[tex]\[
\frac{4}{15} + 0.4 = \frac{4}{15} + \frac{4}{10} = \frac{4}{15} + \frac{6}{15} = \frac{10}{15} = \frac{2}{3}
\][/tex]
4. Now, consider the whole expression again:
[tex]\[
\frac{2}{3} - 1 = \frac{2}{3} - \frac{3}{3} = -\frac{1}{3}
\][/tex]
So, the value of the expression is [tex]\(\boxed{-\frac{1}{3}}\)[/tex], which corresponds to [tex]\(-0.33333...\)[/tex].
Case 2: [tex]\( x = -\frac{4}{9} \)[/tex]
1. Substitute [tex]\( x = -\frac{4}{9} \)[/tex] into the expression:
[tex]\[
\frac{4}{15} - \frac{4}{9} + 0.4
\][/tex]
2. Simplify the expression step by step:
[tex]\[
\frac{4}{15} - \frac{4}{9} = \frac{4}{15} - \frac{20}{45} = \frac{12}{45} - \frac{20}{45} = -\frac{8}{45}
\][/tex]
3. Combine this with 0.4:
[tex]\[
-\frac{8}{45} + 0.4
\][/tex]
4. Convert 0.4 to a fraction with the same denominator:
[tex]\[
0.4 = \frac{4}{10} = \frac{18}{45}
\][/tex]
5. Add the fractions:
[tex]\[
-\frac{8}{45} + \frac{18}{45} = \frac{10}{45} = \frac{2}{9}
\][/tex]
So, the value of the expression is [tex]\(\boxed{\frac{2}{9}}\)[/tex], which corresponds to approximately [tex]\(0.22222...\)[/tex].
Case 3: [tex]\( x = 1 \frac{1}{3} \)[/tex]
1. Convert [tex]\(x\)[/tex] to an improper fraction: [tex]\( 1 \frac{1}{3} = \frac{4}{3} \)[/tex]
2. Substitute [tex]\( x = \frac{4}{3} \)[/tex] into the expression:
[tex]\[
\frac{4}{15} + \frac{4}{3} + 0.4
\][/tex]
3. Simplify the expression step by step:
[tex]\[
\frac{4}{15} + \frac{20}{15} = \frac{24}{15}= \frac{8}{5}
\][/tex]
4. Convert 0.4 to a fraction:
[tex]\[
0.4 = \frac{2}{5}
\][/tex]
5. Add the fractions:
[tex]\[
\frac{8}{5} + \frac{2}{5} = \frac{10}{5} = 2
\][/tex]
So, the value of the expression is [tex]\(\boxed{2}\)[/tex].
