# 004 Sample Final A, Problem 3

Solve. Provide your solution in interval notation.     ${\displaystyle \vert 4x+7\vert \geq 5}$

Foundations
1) What is the solution to ${\displaystyle |x|\geq 3}$?
2) How do you write ${\displaystyle x\geq 2}$ in interval notation?
1) The solution is ${\displaystyle x\geq 3}$ or ${\displaystyle x\leq -3}$.
2) ${\displaystyle [2,\infty )}$

Solution:

Step 1:
The inequality above means ${\displaystyle 4x+7\geq 5}$ or ${\displaystyle 4x+7\leq -5}$.
Step 2:
Subtracting 7 from both sides of the inequalities, we get ${\displaystyle 4x\geq -2}$ or ${\displaystyle 4x\leq -12}$.
Step 3:
Dividing both sides of the inequalities by 4, we have ${\displaystyle x\geq -{\frac {1}{2}}}$ or ${\displaystyle x\leq -3}$.
Step 4:
Using interval notation, the solution is ${\displaystyle (-\infty ,-3]\cup [-{\frac {1}{2}},\infty )}$.
${\displaystyle (-\infty ,-3]\cup [-{\frac {1}{2}},\infty )}$