Questions: Solve the given inequality. (4a+5)/3 - (5a-7)/4 > a+8 Enter the solution set in interval notation:

Solution Steps

To solve the given inequality, we first need to find a common denominator for the fractions on the left side of the inequality. Once we have a common denominator, we can combine the fractions into a single expression. Next, we will move all terms involving the variable \(a\) to one side of the inequality and constant terms to the other side. Finally, we will solve for \(a\) and express the solution in interval notation.

To solve the inequality \(\frac{4a+5}{3} - \frac{5a-7}{4} > a + 8\), we first find a common denominator for the fractions on the left side. The common denominator for 3 and 4 is 12. Rewriting the fractions, we have:

\[ \frac{16a + 20}{12} - \frac{15a - 21}{12} > a + 8 \]

Combine the fractions:

\[ \frac{16a + 20 - (15a - 21)}{12} > a + 8 \]

Simplify the numerator:

\[ \frac{16a + 20 - 15a + 21}{12} = \frac{a + 41}{12} \]

So the inequality becomes:

\[ \frac{a + 41}{12} > a + 8 \]

Multiply both sides by 12 to eliminate the fraction:

\[ a + 41 > 12a + 96 \]

Rearrange the terms to isolate \(a\):

\[ 41 - 96 > 12a - a \]

\[ -55 > 11a \]

Divide both sides by 11:

\[ -\frac{55}{11} > a \]

\[ -5 > a \]

Final Answer