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To determine the number of integers that satisfy the inequality [tex]\((x^2 - 4)(x^2 - 10) < 0\)[/tex], we can follow these steps:
1. Understand the Inequality:
The inequality given is [tex]\((x^2 - 4)(x^2 - 10) < 0\)[/tex]. This inequality involves a polynomial expression.
2. Factor the Polynomial:
Note that [tex]\( x^2 - 4 \)[/tex] can be factored as [tex]\((x - 2)(x + 2)\)[/tex]. So, the inequality becomes:
[tex]\[ (x - 2)(x + 2)(x^2 - 10) < 0 \][/tex]
3. Find the Critical Points:
The factors [tex]\( x - 2 \)[/tex], [tex]\( x + 2 \)[/tex], and [tex]\( x^2 - 10 \)[/tex] are equal to zero at the points [tex]\( x = 2 \)[/tex], [tex]\( x = -2 \)[/tex], and [tex]\( x = \pm\sqrt{10} \)[/tex] respectively. For simplicity:
[tex]\[ x = -\sqrt{10}, -2, 2, \sqrt{10} \][/tex]
4. Determine the Intervals:
These critical points divide the number line into several intervals:
[tex]\[ (-\infty, -\sqrt{10}), (-\sqrt{10}, -2), (-2, 2), (2, \sqrt{10}), (\sqrt{10}, \infty) \][/tex]
5. Test Points in Each Interval:
We need to test points in each interval to determine where the product [tex]\((x - 2)(x + 2)(x^2 - 10)\)[/tex] is negative.
- Interval [tex]\((- \infty, -\sqrt{10})\)[/tex]: Choose [tex]\(x = -4\)[/tex], and test the sign of [tex]\((x - 2)(x + 2)(x^2 - 10)\)[/tex].
- Interval [tex]\((- \sqrt{10}, -2)\)[/tex]: Choose [tex]\(x = -3\)[/tex], and test the sign.
- Interval [tex]\((-2, 2)\)[/tex]: Choose [tex]\(x = 0\)[/tex], and test the sign.
- Interval [tex]\((2, \sqrt{10})\)[/tex]: Choose [tex]\(x = 3\)[/tex], and test the sign.
- Interval [tex]\((\sqrt{10}, \infty)\)[/tex]: Choose [tex]\(x = 4\)[/tex], and test the sign.
6. Verify the Signs:
From these tests, it will be observed that:
- The inequality is satisfied in the interval [tex]\((- \sqrt{10}, -2)\)[/tex] and [tex]\((2, \sqrt{10})\)[/tex].
7. Identify the Integer Solutions:
Within these intervals, we identify the integer values:
[tex]\[ \text{From } (- \sqrt{10}, -2)\text{, the integers are } -3. \][/tex]
[tex]\[ \text{From } (2, \sqrt{10})\text{, the integers are } 3. \][/tex]
Therefore, the integers that satisfy the inequality are [tex]\(-3\)[/tex] and [tex]\(3\)[/tex].
8. Count the Integer Solutions:
The number of integers that satisfy the inequality is:
[tex]\[ 2 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
1. Understand the Inequality:
The inequality given is [tex]\((x^2 - 4)(x^2 - 10) < 0\)[/tex]. This inequality involves a polynomial expression.
2. Factor the Polynomial:
Note that [tex]\( x^2 - 4 \)[/tex] can be factored as [tex]\((x - 2)(x + 2)\)[/tex]. So, the inequality becomes:
[tex]\[ (x - 2)(x + 2)(x^2 - 10) < 0 \][/tex]
3. Find the Critical Points:
The factors [tex]\( x - 2 \)[/tex], [tex]\( x + 2 \)[/tex], and [tex]\( x^2 - 10 \)[/tex] are equal to zero at the points [tex]\( x = 2 \)[/tex], [tex]\( x = -2 \)[/tex], and [tex]\( x = \pm\sqrt{10} \)[/tex] respectively. For simplicity:
[tex]\[ x = -\sqrt{10}, -2, 2, \sqrt{10} \][/tex]
4. Determine the Intervals:
These critical points divide the number line into several intervals:
[tex]\[ (-\infty, -\sqrt{10}), (-\sqrt{10}, -2), (-2, 2), (2, \sqrt{10}), (\sqrt{10}, \infty) \][/tex]
5. Test Points in Each Interval:
We need to test points in each interval to determine where the product [tex]\((x - 2)(x + 2)(x^2 - 10)\)[/tex] is negative.
- Interval [tex]\((- \infty, -\sqrt{10})\)[/tex]: Choose [tex]\(x = -4\)[/tex], and test the sign of [tex]\((x - 2)(x + 2)(x^2 - 10)\)[/tex].
- Interval [tex]\((- \sqrt{10}, -2)\)[/tex]: Choose [tex]\(x = -3\)[/tex], and test the sign.
- Interval [tex]\((-2, 2)\)[/tex]: Choose [tex]\(x = 0\)[/tex], and test the sign.
- Interval [tex]\((2, \sqrt{10})\)[/tex]: Choose [tex]\(x = 3\)[/tex], and test the sign.
- Interval [tex]\((\sqrt{10}, \infty)\)[/tex]: Choose [tex]\(x = 4\)[/tex], and test the sign.
6. Verify the Signs:
From these tests, it will be observed that:
- The inequality is satisfied in the interval [tex]\((- \sqrt{10}, -2)\)[/tex] and [tex]\((2, \sqrt{10})\)[/tex].
7. Identify the Integer Solutions:
Within these intervals, we identify the integer values:
[tex]\[ \text{From } (- \sqrt{10}, -2)\text{, the integers are } -3. \][/tex]
[tex]\[ \text{From } (2, \sqrt{10})\text{, the integers are } 3. \][/tex]
Therefore, the integers that satisfy the inequality are [tex]\(-3\)[/tex] and [tex]\(3\)[/tex].
8. Count the Integer Solutions:
The number of integers that satisfy the inequality is:
[tex]\[ 2 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
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