Certainly! Let's solve the given inequality step-by-step.
Given:
[tex]\[
2r - \frac{5}{3}r > -\frac{1}{3}
\][/tex]
Step 1: Combine like terms on the left-hand side.
First, write the term [tex]\(2r\)[/tex] with a common denominator of 3:
[tex]\[
2r = \frac{6}{3}r
\][/tex]
Now we can rewrite the inequality, substituting [tex]\(2r\)[/tex]:
[tex]\[
\frac{6}{3}r - \frac{5}{3}r > -\frac{1}{3}
\][/tex]
Step 2: Simplify the left side by combining the fractions:
Subtract the fractions:
[tex]\[
\left(\frac{6}{3}r - \frac{5}{3}r\right) = \frac{6 - 5}{3}r = \frac{1}{3}r
\][/tex]
So the inequality is now:
[tex]\[
\frac{1}{3}r > -\frac{1}{3}
\][/tex]
Step 3: Isolate [tex]\(r\)[/tex] by eliminating the fraction:
To get rid of the denominator, multiply both sides of the inequality by 3:
[tex]\[
3 \cdot \frac{1}{3}r > 3 \cdot -\frac{1}{3}
\][/tex]
Which simplifies to:
[tex]\[
r > -1
\][/tex]
Thus, the simplified inequality is:
[tex]\[
r > -1
\][/tex]
