004 Sample Final A, Problem 3

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

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

Solution:

Step 1:
The inequality above means $4x+7\geq 5$ or $4x+7\leq -5$.

Step 2:
Subtracting 7 from both sides of the inequalities, we get $4x\geq -2$ or $4x\leq -12$.

Step 3:
Dividing both sides of the inequalities by 4, we have $x\geq -{\frac {1}{2}}$ or $x\leq -3$.

Step 4:
Using interval notation, the solution is $(-\infty ,-3]\cup [-{\frac {1}{2}},\infty )$.

Final Answer:
$(-\infty ,-3]\cup [-{\frac {1}{2}},\infty )$

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