QuestionSolve the inequality and graph the solution on the real number line. $(x-4)^{2} \geq 1$

Studdy Solution

STEP 1

What is this asking? Find all the $x$ values that make $(x-4)^2$ greater than or equal to 1, and then show those values on a number line. Watch out! Don't forget to consider both the positive and negative square roots when solving inequalities involving squares.

STEP 2

1. Rewrite the inequality.
2. Solve for x.
3. Graph the solution.

STEP 3

We're dealing with a squared expression greater than or equal to a number.
This is like saying the distance between $x$ and 4, squared, is at least 1.
Let's rewrite the inequality to make it easier to solve.
We can think of this as the difference of squares!

STEP 4

$(x-4)^2 - 1 \geq 0$ We subtracted 1 from both sides.
Why? Because this sets us up to use the difference of squares!

STEP 5

$((x-4) - 1)((x-4) + 1) \geq 0$ Here, we used the difference of squares factorization: $a^2 - b^2 = (a-b)(a+b)$ .
This makes things much easier to handle!

STEP 6

$(x-5)(x-3) \geq 0$ We simplified the factors.
Now we have a much cleaner inequality!

STEP 7

Now, we need to figure out when $(x-5)(x-3)$ is greater than or equal to zero.
This happens when both factors are positive, or both factors are negative, or when one of the factors is zero.

STEP 8

Case 1: Both factors are positive $x - 5 \geq 0$ and $x - 3 \geq 0$ .
This means $x \geq 5$ and $x \geq 3$ .
Since x has to be greater than or equal to both 5 and 3, we take the larger value.
So, $x \geq 5$ .

STEP 9

Case 2: Both factors are negative $x - 5 \leq 0$ and $x - 3 \leq 0$ .
This means $x \leq 5$ and $x \leq 3$ .
Since x has to be less than or equal to both 5 and 3, we take the smaller value.
So, $x \leq 3$ .

STEP 10

Case 3: One factor is zero If $x - 5 = 0$ , then $x = 5$ .
If $x - 3 = 0$ , then $x = 3$ .

STEP 11

Combining all cases, our solution is $x \leq 3$ or $x \geq 5$ .

STEP 12

On a number line, we'll represent the solution.
We'll use filled-in circles at $x=3$ and $x=5$ because the inequality includes "equal to." Then, we shade the line to the left of 3 and to the right of 5.

STEP 13

The solution to the inequality $(x-4)^2 \geq 1$ is $x \leq 3$ or $x \geq 5$ .
This is represented on the number line with closed circles at 3 and 5, and shaded regions extending to the left of 3 and to the right of 5.

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