QuestionSolve the inequality: $(x-3)(x-4)(x-7) \leq 0$ . Provide the solution in interval notation.

Studdy Solution

STEP 1

Assumptions1. The inequality is $(x-3)(x-4)(x-7) \leq0$ . We need to find the values of $x$ for which this inequality holds true

STEP 2

To solve the inequality, we first need to find the critical points. These are the values of $x$ that make the expression equal to zero.
$(x-)(x-4)(x-7) =0$

STEP 3

Setting each factor equal to zero gives us the critical points.
$x-3=0, x-=0, x-7=0$

STEP 4

olving each equation gives us the critical points.
$x=3, x=4, x=7$

STEP 5

These critical points divide the number line into four intervals. We will test each interval to see where the inequality holds true.
The intervals are $(-\infty,3)$ , $(3,4)$ , $(4,7)$ , and $(7, \infty)$ .

STEP 6

Choose a test point in the interval $(-\infty,3)$ , let's choose $x=0$ .
Substitute $x=0$ into the inequality $(0-3)(0-4)(0-) \leq0$

STEP 7

Calculate the result of the inequality for $x=0$ .
$(-3)(-4)(-7) \leq0$

STEP 8

The result is $84 \leq0$ , which is false. So, the interval $(-\infty,3)$ is not part of the solution.

STEP 9

Choose a test point in the interval $(3,4)$ , let's choose $x=3.5$ .
Substitute $x=3.5$ into the inequality $(3.5-3)(3.5-4)(3.5-7) \leq$

STEP 10

Calculate the result of the inequality for $x=3.5$ .
$(0.5)(-0.5)(-3.5) \leq0$

STEP 11

The result is $0.875 \leq0$ , which is false. So, the interval $(3,4)$ is not part of the solution.

STEP 12

Choose a test point in the interval $(4,7)$ , let's choose $x=5$ .
Substitute $x=5$ into the inequality $(5-)(5-4)(5-7) \leq0$

STEP 13

Calculate the result of the inequality for $x=5$ .
$(2)()(-2) \leq0$

STEP 14

The result is $-4 \leq0$ , which is true. So, the interval $(4,7)$ is part of the solution.

STEP 15

Choose a test point in the interval $(7, \infty)$ , let's choose $x=8$ .
Substitute $x=8$ into the inequality $(8-3)(8-4)(8-7) \leq0$

STEP 16

Calculate the result of the inequality for $x=8$ .
$(5)(4)() \leq0$

STEP 17

The result is $20 \leq0$ , which is false. So, the interval $(7, \infty)$ is not part of the solution.

STEP 18

The inequality holds true for the interval $(4,7)$ and the critical points $x=3$ , $x=4$ , and $x=7$ . Therefore, the solution to the inequality is $[3,4] \cup (4,7]$

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