Let's solve for the input [tex]\( x \)[/tex] that gives an output value of [tex]\( \frac{2}{3} \)[/tex] in the function [tex]\( f(x) = -\frac{1}{3}x + 7 \)[/tex].
1. Start with the given function and set it equal to [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[ \frac{2}{3} = -\frac{1}{3}x + 7 \][/tex]
2. To isolate the term with [tex]\( x \)[/tex], subtract 7 from both sides of the equation:
[tex]\[ \frac{2}{3} - 7 = -\frac{1}{3}x \][/tex]
3. Simplify the left side:
[tex]\[ \frac{2}{3} - 7 = \frac{2}{3} - \frac{21}{3} = \frac{2 - 21}{3} = -\frac{19}{3} \][/tex]
4. Now, we have:
[tex]\[ -\frac{19}{3} = -\frac{1}{3}x \][/tex]
5. To solve for [tex]\( x \)[/tex], multiply both sides of the equation by [tex]\(-3\)[/tex]:
[tex]\[ x = \left(-\frac{19}{3}\right) \times (-3) \][/tex]
6. Simplifying this, we get:
[tex]\[ x = 19 \][/tex]
Therefore, the input [tex]\( x \)[/tex] that results in an output value of [tex]\( \frac{2}{3} \)[/tex] is [tex]\( x = 19 \)[/tex].
