Questions: Solve the compound inequality. -3u < -18 and 4u - 5 > -29 Write the solution in interval notation. If there is no solution, enter ∅.

Transcript text: Solve the compound inequality. $-3 u<-18$ and $4 u-5>-29$ Write the solution in interval notation. If there is no solution, enter $\varnothing$.

Solution

Solution Steps

To solve the compound inequality, we need to solve each inequality separately and then find the intersection of the solutions. For the first inequality, $-3u < -18$, we will isolate $u$ by dividing both sides by -3, remembering to reverse the inequality sign. For the second inequality, $4u - 5 > -29$, we will first add 5 to both sides and then divide by 4 to isolate $u$. Finally, we will find the intersection of the two solution sets and express it in interval notation.

Step 1: Solve the First Inequality

The first inequality is:

\[ -3u < -18 \]

To solve for $ u $, divide both sides by $-3$:

\[ u > \frac{-18}{-3} \]

\[ u > 6 \]

Step 2: Solve the Second Inequality

The second inequality is:

\[ 4u - 5 > -29 \]

First, add 5 to both sides:

\[ 4u > -29 + 5 \]

\[ 4u > -24 \]

Next, divide both sides by 4:

\[ u > \frac{-24}{4} \]

\[ u > -6 \]

Step 3: Combine the Solutions

We have two inequalities:

$ u > 6 $
$ u > -6 $

The solution to the compound inequality is the intersection of these two solutions. Since $ u > 6 $ is more restrictive than $ u > -6 $, the solution is:

\[ u > 6 \]