To solve the inequality [tex]\( 7 + 2x \geq -5 - x \)[/tex], we need to isolate the variable [tex]\( x \)[/tex]. Here’s the step-by-step solution:
1. Start with the given inequality:
[tex]\[
7 + 2x \geq -5 - x
\][/tex]
2. Move all terms involving [tex]\( x \)[/tex] to one side of the inequality:
To do this, we can add [tex]\( x \)[/tex] to both sides:
[tex]\[
7 + 2x + x \geq -5 - x + x
\][/tex]
Simplifying this, we get:
[tex]\[
7 + 3x \geq -5
\][/tex]
3. Move the constant terms to the other side of the inequality:
To isolate the term involving [tex]\( x \)[/tex], we subtract 7 from both sides:
[tex]\[
7 + 3x - 7 \geq -5 - 7
\][/tex]
Simplifying this, we get:
[tex]\[
3x \geq -12
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides of the inequality by 3:
[tex]\[
\frac{3x}{3} \geq \frac{-12}{3}
\][/tex]
Simplifying the division, we obtain:
[tex]\[
x \geq -4
\][/tex]
Thus, the solution to the inequality [tex]\( 7 + 2x \geq -5 - x \)[/tex] is:
[tex]\[
x \geq -4
\][/tex]
This can be written in interval notation as:
[tex]\[
[-4, \infty)
\][/tex]
We have thus determined that [tex]\( x \)[/tex] lies in the interval [tex]\([-4, \infty)\)[/tex].
