Certainly! Let's solve the given compound inequality step-by-step:
[tex]\[ x + 6 < -3x \quad \text{and} \quad x < 2 \][/tex]
### Step 1: Solve the first inequality
First, let's isolate [tex]\( x \)[/tex] in the inequality [tex]\( x + 6 < -3x \)[/tex].
1. Add [tex]\( 3x \)[/tex] to both sides:
[tex]\[
x + 6 + 3x < -3x + 3x
\][/tex]
[tex]\[
4x + 6 < 0
\][/tex]
2. Subtract 6 from both sides:
[tex]\[
4x + 6 - 6 < 0 - 6
\][/tex]
[tex]\[
4x < -6
\][/tex]
3. Divide both sides by 4:
[tex]\[
\frac{4x}{4} < \frac{-6}{4}
\][/tex]
[tex]\[
x < -\frac{3}{2}
\][/tex]
So, the solution to the first inequality is:
[tex]\[ x < -\frac{3}{2} \][/tex]
### Step 2: Solve the second inequality
The second inequality is already in a simple form:
[tex]\[ x < 2 \][/tex]
### Step 3: Combine the solutions
Since the inequalities must both be true simultaneously, we need to find the intersection of the two solution sets. The solution to [tex]\( x < -\frac{3}{2} \)[/tex] is all the values less than [tex]\( -\frac{3}{2} \)[/tex], and the solution to [tex]\( x < 2 \)[/tex] is all the values less than [tex]\( 2 \)[/tex].
The more restrictive condition is [tex]\( x < -\frac{3}{2} \)[/tex], because any value that meets this condition will also meet [tex]\( x < 2 \)[/tex].
### Step 4: Write the final solution
Therefore, combining both inequalities, the solution to the compound inequality is:
[tex]\[ x < -\frac{3}{2} \][/tex]
This conclusion represents the values of [tex]\( x \)[/tex] that satisfy both inequalities simultaneously.
