Questions: Solve the compound inequality. -3u > -18 and 4u - 2 ≤ -18 Write the solution in interval notation. If there is no solution, enter ∅.

Transcript text: Solve the compound inequality. \[ -3 u>-18 \text { and } 4 u-2 \leq-18 \] Write the solution in interval notation. If there is no solution, enter $\varnothing$. $\square$

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, $4u - 2 \leq -18$, we will first add 2 to both sides and then divide by 4 to solve for $u$. 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 given is:

\[ -3u > -18 \]

To solve for $ u $, divide both sides by $-3$. Remember that dividing by a negative number reverses the inequality sign:

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

\[ u < 6 \]

Step 2: Solve the Second Inequality

The second inequality given is:

\[ 4u - 2 \leq -18 \]

First, add 2 to both sides to isolate the term with $ u $:

\[ 4u \leq -18 + 2 \]

\[ 4u \leq -16 \]

Next, divide both sides by 4:

\[ u \leq \frac{-16}{4} \]

\[ u \leq -4 \]

Step 3: Combine the Solutions

We have two inequalities:

$ u < 6 $
$ u \leq -4 $

The solution to the compound inequality is the intersection of these two solutions. The intersection is the set of values that satisfy both inequalities simultaneously.

Since $ u \leq -4 $ is more restrictive than $ u < 6 $ for values less than 6, the solution is:

\[ u \leq -4 \]