Solve the following inequalities and illustrate the results on a number line: x² > 16, x² < 25, x² - 3x - 4 > 0. Also find the solution of the following inequalities: (x-2)(x+3) >... Solve the following inequalities and illustrate the results on a number line: x² > 16, x² < 25, x² - 3x - 4 > 0. Also find the solution of the following inequalities: (x-2)(x+3) > 0, x² - 7x + 10 < 0, and x² + x - 12 > 0.
Understand the Problem
The question involves solving multiple inequalities and illustrating the results on a number line, including factoring and solving quadratic inequalities.
Answer
$$ x < -3 \quad \text{or} \quad x > 4 $$
Answer for screen readers
The solution to the inequalities is: $$ x < -3 \quad \text{or} \quad x > 4 $$
Steps to Solve
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Solve the first inequality ( x^2 - 3x - 4 > 0 )
First, we need to find the roots of the quadratic equation ( x^2 - 3x - 4 = 0 ) using the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Here, ( a = 1 ), ( b = -3 ), and ( c = -4 ).
Calculate the discriminant: $$ b^2 - 4ac = (-3)^2 - 4(1)(-4) = 9 + 16 = 25 $$
Now, substitute back into the quadratic formula: $$ x = \frac{3 \pm 5}{2} $$ This gives: $$ x = 4 \quad \text{and} \quad x = -1 $$
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Test intervals for the inequality
The critical numbers divide the number line into intervals: ( (-\infty, -1) ), ( (-1, 4) ), and ( (4, \infty) ).
Choose test points in each interval:
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For interval ( (-\infty, -1) ): Choose ( x = -2 )
- ( (-2)^2 - 3(-2) - 4 = 4 + 6 - 4 = 6 > 0 ) (True)
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For interval ( (-1, 4) ): Choose ( x = 0 )
- ( 0^2 - 3(0) - 4 = -4 < 0 ) (False)
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For interval ( (4, \infty) ): Choose ( x = 5 )
- ( 5^2 - 3(5) - 4 = 25 - 15 - 4 = 6 > 0 ) (True)
Therefore, the solution is: $$ x < -1 \quad \text{or} \quad x > 4 $$
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Solve the second inequality ( (x - 2)(x + 3) > 0 )
Find the roots by solving: $$ (x - 2)(x + 3) = 0 $$ This gives: $$ x = 2 \quad \text{and} \quad x = -3 $$
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Test intervals for the second inequality
The critical points divide the number line into intervals: ( (-\infty, -3) ), ( (-3, 2) ), and ( (2, \infty) ).
Choose test points in each interval:
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For interval ( (-\infty, -3) ): Choose ( x = -4 )
- ( (-4 - 2)(-4 + 3) = (-6)(-1) = 6 > 0 ) (True)
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For interval ( (-3, 2) ): Choose ( x = 0 )
- ( (0 - 2)(0 + 3) = (-2)(3) = -6 < 0 ) (False)
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For interval ( (2, \infty) ): Choose ( x = 3 )
- ( (3 - 2)(3 + 3) = (1)(6) = 6 > 0 ) (True)
Therefore, the solution is: $$ x < -3 \quad \text{or} \quad x > 2 $$
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Combine solutions on a number line
We have two inequalities:
- From the first: ( x < -1 \quad \text{or} \quad x > 4 )
- From the second: ( x < -3 \quad \text{or} \quad x > 2 )
The combined solution indicates where both inequalities hold.
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Finalize the solution
The final solution combines all tested intervals:
- ( x < -3 \quad \text{or} \quad x > 4 )
The solution to the inequalities is: $$ x < -3 \quad \text{or} \quad x > 4 $$
More Information
This result means that the valid ( x ) values satisfying both inequalities exist in two regions of the number line.
Tips
- Ignoring test points: Always choose a point from each interval to test the inequality.
- Confusing the inequality sign: Remember to accurately interpret the result from testing each interval.
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