Certainly! Let's solve the inequality step by step.
The inequality we need to solve is:
[tex]\[ 4x + 16 > -4(x + 3) - 2x \][/tex]
1. Distribute and simplify both sides:
First, let's distribute the [tex]\(-4\)[/tex] on the right side:
[tex]\[ -4(x + 3) = -4x - 12 \][/tex]
Thus, the inequality becomes:
[tex]\[ 4x + 16 > -4x - 12 - 2x \][/tex]
2. Combine like terms on the right side:
Now simplify the right side by combining the [tex]\(x\)[/tex] terms:
[tex]\[ -4x - 2x = -6x \][/tex]
So the inequality is now:
[tex]\[ 4x + 16 > -6x - 12 \][/tex]
3. Move all [tex]\(x\)[/tex]-terms to one side and constant terms to the other side:
Add [tex]\(6x\)[/tex] to both sides:
[tex]\[ 4x + 6x + 16 > -12 \][/tex]
[tex]\[ 10x + 16 > -12 \][/tex]
Next, subtract 16 from both sides:
[tex]\[ 10x > -12 - 16 \][/tex]
[tex]\[ 10x > -28 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 10:
[tex]\[ x > \frac{-28}{10} \][/tex]
[tex]\[ x > -\frac{14}{5} \][/tex]
Therefore, the solution to the inequality [tex]\(4x + 16 > -4(x + 3) - 2x\)[/tex] is:
[tex]\[ x > -\frac{14}{5} \][/tex]
In interval notation, this is expressed as:
[tex]\[ \left( -\frac{14}{5}, \infty \right) \][/tex]
Summary:
[tex]\[ 4x + 16 > -4(x + 3) - 2x \][/tex]
[tex]\[ x > -\frac{14}{5} \][/tex]
[tex]\[ \left( -\frac{14}{5}, \infty \right) \][/tex]
