To solve the inequality [tex]\( x - 2 \leq 4x + 7 \)[/tex], we need to isolate the variable [tex]\( x \)[/tex]. Here are the steps:
1. Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[
x - 2 - x \leq 4x + 7 - x
\][/tex]
Simplifying the terms, we get:
[tex]\[
-2 \leq 3x + 7
\][/tex]
2. Subtract 7 from both sides to further isolate [tex]\( x \)[/tex]:
[tex]\[
-2 - 7 \leq 3x + 7 - 7
\][/tex]
Simplifying the terms, we get:
[tex]\[
-9 \leq 3x
\][/tex]
3. Divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{-9}{3} \leq \frac{3x}{3}
\][/tex]
Simplifying the terms, we get:
[tex]\[
-3 \leq x
\][/tex]
So, the solution to the inequality is [tex]\( x \geq -3 \)[/tex].
To write this solution set in interval notation, we denote all [tex]\( x \)[/tex] values that are greater than or equal to [tex]\(-3\)[/tex], which includes everything from [tex]\(-3\)[/tex] to positive infinity. In interval notation, this is written as:
[tex]\[
[-3, \infty)
\][/tex]
This means [tex]\( x \)[/tex] can take any value from [tex]\(-3\)[/tex] inclusive, and extend to infinity.
