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Help I dont know what to do her

 

Solve the inequality
\frac{3-z}{z+1} \ge 2(z + 4).
Write your answer in interval notation.

 Dec 18, 2023
 #1
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Here's how to solve the inequality and express the answer in interval notation:

 

1. Simplify the inequality:

 

Multiply both sides by z+1:

 

3-z >= 2(z+1)(z+1) 3-z >= 2z^2 + 10z + 8

 

Combine like terms:

 

0 >= 2z^2 + 11z + 5

 

2. Factor the quadratic: The quadratic expression cannot be factored into simpler terms, so we move on to analyzing its sign changes.

 

3. Find the roots and analyze sign changes:

 

Solve the quadratic equation for the roots:

 

2z^2 + 11z + 5 = 0 z = (-11 ± sqrt(11^2 - 4*2*5)) / (2*2) z = (-11 ± sqrt(91)) / 4 z = (-11 ± √91) / 4 z ≈ -1.75, -6.25

 

Note the roots and create three intervals on the number line: | Interval | Left | Right | |---|---|---| | z < -6.25 | | X | -6.25 | | -6.25 <= z <= -1.75 | X | -6.25 | -1.75 | | z > -1.75 | -1.75 | X |

 

4. Evaluate the inequality in each interval:

 

Choose a value within each interval and evaluate the inequality using that value:

 

 | Interval | z-value | Evaluation | |---|---|---| | z < -6.25 | z = -7 | (-3+7)/(7+1) > 2(-7+4) (True) | | -6.25 <= z <= -1.75 | z = -4 | (-3+4)/(-4+1) > 2(-4+4) (False) | | z > -1.75 | z = 0 | (-3+0)/(0+1) > 2(0+4) (False) |

 

5. Identify valid intervals: Based on the evaluations, the inequality holds true only in the interval where z is less than -6.25.

 

6. Write the solution in interval notation: Therefore, the solution to the inequality is:

 

z < -6.25

 

This represents the set of all real numbers less than -6.25.

 Dec 18, 2023

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