Solved on Jan 23, 2024

Find the solution set for the inequality $(x-2)(x+1)(x-4) \geq 0$ . The solution set is $[-1,2],[4, \infty)$ .

STEP 1

Assumptions
1. We are given the inequality $(x-2)(x+1)(x-4) \geq 0$ .
2. We need to find the solution set for $x$ that satisfies this inequality.
3. The inequality is a product of three linear factors.

STEP 2

Identify the critical points of the inequality by setting each factor to zero.
$x - 2 = 0, \quad x + 1 = 0, \quad x - 4 = 0$

STEP 3

Solve each equation for $x$ to find the critical points.
$x = 2, \quad x = -1, \quad x = 4$

STEP 4

Plot the critical points on a number line and determine the intervals they create.
The critical points divide the number line into four intervals: $(-\infty, -1), \quad (-1, 2), \quad (2, 4), \quad (4, \infty)$

STEP 5

Determine the sign of the product $(x-2)(x+1)(x-4)$ in each interval by choosing a test point from each interval and plugging it into the inequality.

STEP 6

Choose test points for each interval. A good choice of test points is: $x = -2 \text{ for } (-\infty, -1)$ $x = 0 \text{ for } (-1, 2)$ $x = 3 \text{ for } (2, 4)$ $x = 5 \text{ for } (4, \infty)$

STEP 7

Evaluate the sign of the product at $x = -2$ .
$(x-2)(x+1)(x-4) = (-2-2)(-2+1)(-2-4) = (-4)(-1)(-6)$

STEP 8

Determine the sign of the product at $x = -2$ .
$(-4)(-1)(-6) < 0$

STEP 9

Evaluate the sign of the product at $x = 0$ .
$(x-2)(x+1)(x-4) = (0-2)(0+1)(0-4) = (-2)(1)(-4)$

STEP 10

Determine the sign of the product at $x = 0$ .
$(-2)(1)(-4) > 0$

STEP 11

Evaluate the sign of the product at $x = 3$ .
$(x-2)(x+1)(x-4) = (3-2)(3+1)(3-4) = (1)(4)(-1)$

STEP 12

Determine the sign of the product at $x = 3$ .
$(1)(4)(-1) < 0$

STEP 13

Evaluate the sign of the product at $x = 5$ .
$(x-2)(x+1)(x-4) = (5-2)(5+1)(5-4) = (3)(6)(1)$

STEP 14

Determine the sign of the product at $x = 5$ .
$(3)(6)(1) > 0$

STEP 15

Using the test points, we can determine the sign of the product in each interval:
For $(-\infty, -1)$ , the product is negative. For $(-1, 2)$ , the product is positive. For $(2, 4)$ , the product is negative. For $(4, \infty)$ , the product is positive.

STEP 16

Since we are looking for the intervals where the product is greater than or equal to zero, we select the intervals where the product is positive or zero at the endpoints.

STEP 17

The intervals where the product is positive are $(-1, 2)$ and $(4, \infty)$ . We also need to include the endpoints where the product is zero, which are $x = -1$ , $x = 2$ , and $x = 4$ .

STEP 18

Combine the intervals and endpoints to write the solution set.
The solution set is $[-1, 2] \cup [4, \infty)$ .

STEP 19

Compare the solution set with the given options.
The correct option that matches our solution set is: $[-1,2],[4, \infty)$

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Find the solution set for the inequality (x−2)(x+1)(x−4)≥0(x-2)(x+1)(x-4) \geq 0(x−2)(x+1)(x−4)≥0. The solution set is [−1,2],[4,∞)[-1,2],[4, \infty)[−1,2],[4,∞).

STEP 1

Assumptions1. We are given the inequality (x−2)(x+1)(x−4)≥0(x-2)(x+1)(x-4) \geq 0(x−2)(x+1)(x−4)≥0.2. We need to find the solution set for xxx that satisfies this inequality.3. The inequality is a product of three linear factors.

STEP 2

Identify the critical points of the inequality by setting each factor to zero.x−2=0,x+1=0,x−4=0x - 2 = 0, \quad x + 1 = 0, \quad x - 4 = 0x−2=0,x+1=0,x−4=0

STEP 3

Solve each equation for xxx to find the critical points.x=2,x=−1,x=4x = 2, \quad x = -1, \quad x = 4x=2,x=−1,x=4

STEP 4

Plot the critical points on a number line and determine the intervals they create.The critical points divide the number line into four intervals: (−∞,−1),(−1,2),(2,4),(4,∞)(-\infty, -1), \quad (-1, 2), \quad (2, 4), \quad (4, \infty)(−∞,−1),(−1,2),(2,4),(4,∞)

STEP 5

Determine the sign of the product (x−2)(x+1)(x−4)(x-2)(x+1)(x-4)(x−2)(x+1)(x−4) in each interval by choosing a test point from each interval and plugging it into the inequality.

STEP 6

STEP 7

Evaluate the sign of the product at x=−2x = -2x=−2.(x−2)(x+1)(x−4)=(−2−2)(−2+1)(−2−4)=(−4)(−1)(−6)(x-2)(x+1)(x-4) = (-2-2)(-2+1)(-2-4) = (-4)(-1)(-6)(x−2)(x+1)(x−4)=(−2−2)(−2+1)(−2−4)=(−4)(−1)(−6)

STEP 8

Determine the sign of the product at x=−2x = -2x=−2.(−4)(−1)(−6)<0(-4)(-1)(-6) < 0(−4)(−1)(−6)<0

STEP 9

Evaluate the sign of the product at x=0x = 0x=0.(x−2)(x+1)(x−4)=(0−2)(0+1)(0−4)=(−2)(1)(−4)(x-2)(x+1)(x-4) = (0-2)(0+1)(0-4) = (-2)(1)(-4)(x−2)(x+1)(x−4)=(0−2)(0+1)(0−4)=(−2)(1)(−4)

STEP 10

Determine the sign of the product at x=0x = 0x=0.(−2)(1)(−4)>0(-2)(1)(-4) > 0(−2)(1)(−4)>0

STEP 11

Evaluate the sign of the product at x=3x = 3x=3.(x−2)(x+1)(x−4)=(3−2)(3+1)(3−4)=(1)(4)(−1)(x-2)(x+1)(x-4) = (3-2)(3+1)(3-4) = (1)(4)(-1)(x−2)(x+1)(x−4)=(3−2)(3+1)(3−4)=(1)(4)(−1)

STEP 12

Determine the sign of the product at x=3x = 3x=3.(1)(4)(−1)<0(1)(4)(-1) < 0(1)(4)(−1)<0

STEP 13

Evaluate the sign of the product at x=5x = 5x=5.(x−2)(x+1)(x−4)=(5−2)(5+1)(5−4)=(3)(6)(1)(x-2)(x+1)(x-4) = (5-2)(5+1)(5-4) = (3)(6)(1)(x−2)(x+1)(x−4)=(5−2)(5+1)(5−4)=(3)(6)(1)

STEP 14

Determine the sign of the product at x=5x = 5x=5.(3)(6)(1)>0(3)(6)(1) > 0(3)(6)(1)>0

STEP 15

STEP 16

Since we are looking for the intervals where the product is greater than or equal to zero, we select the intervals where the product is positive or zero at the endpoints.

STEP 17

The intervals where the product is positive are (−1,2)(-1, 2)(−1,2) and (4,∞)(4, \infty)(4,∞). We also need to include the endpoints where the product is zero, which are x=−1x = -1x=−1, x=2x = 2x=2, and x=4x = 4x=4.

STEP 18

Combine the intervals and endpoints to write the solution set.The solution set is [−1,2]∪[4,∞)[-1, 2] \cup [4, \infty)[−1,2]∪[4,∞).

STEP 19

Compare the solution set with the given options.The correct option that matches our solution set is: [−1,2],[4,∞)[-1,2],[4, \infty)[−1,2],[4,∞)

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Find the solution set for the inequality $(x-2)(x+1)(x-4) \geq 0$ . The solution set is $[-1,2],[4, \infty)$ .

Assumptions
1. We are given the inequality $(x-2)(x+1)(x-4) \geq 0$ .
2. We need to find the solution set for $x$ that satisfies this inequality.
3. The inequality is a product of three linear factors.

Identify the critical points of the inequality by setting each factor to zero.
$x - 2 = 0, \quad x + 1 = 0, \quad x - 4 = 0$

Solve each equation for $x$ to find the critical points.
$x = 2, \quad x = -1, \quad x = 4$

Plot the critical points on a number line and determine the intervals they create.
The critical points divide the number line into four intervals: $(-\infty, -1), \quad (-1, 2), \quad (2, 4), \quad (4, \infty)$

Determine the sign of the product $(x-2)(x+1)(x-4)$ in each interval by choosing a test point from each interval and plugging it into the inequality.

Evaluate the sign of the product at $x = -2$ .
$(x-2)(x+1)(x-4) = (-2-2)(-2+1)(-2-4) = (-4)(-1)(-6)$

Determine the sign of the product at $x = -2$ .
$(-4)(-1)(-6) < 0$

Evaluate the sign of the product at $x = 0$ .
$(x-2)(x+1)(x-4) = (0-2)(0+1)(0-4) = (-2)(1)(-4)$

Determine the sign of the product at $x = 0$ .
$(-2)(1)(-4) > 0$

Evaluate the sign of the product at $x = 3$ .
$(x-2)(x+1)(x-4) = (3-2)(3+1)(3-4) = (1)(4)(-1)$

Determine the sign of the product at $x = 3$ .
$(1)(4)(-1) < 0$

Evaluate the sign of the product at $x = 5$ .
$(x-2)(x+1)(x-4) = (5-2)(5+1)(5-4) = (3)(6)(1)$

Determine the sign of the product at $x = 5$ .
$(3)(6)(1) > 0$

The intervals where the product is positive are $(-1, 2)$ and $(4, \infty)$ . We also need to include the endpoints where the product is zero, which are $x = -1$ , $x = 2$ , and $x = 4$ .

Combine the intervals and endpoints to write the solution set.
The solution set is $[-1, 2] \cup [4, \infty)$ .

Compare the solution set with the given options.
The correct option that matches our solution set is: $[-1,2],[4, \infty)$