How do you solve and graph #abs(x-3)>=1#?

Answer 1

See a solution process below:

The absolute value function takes any negative or positive term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

#-1 >= x - 3 >= 1#

Add #color(red)(3)# to each segment of the system of inequalities to solve for #x# while keeping the system balanced:

#-1 + color(red)(3) >= x - 3 + color(red)(3) >= 1 + color(red)(3)#

#2 >= x - 0 >= 4#

#2 >= x >= 4#

Or

#x <= 2# and #x >= 4#

Or, in interval notation:

#(-oo, 2]# and #[4, +oo)#

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Answer 2

To solve and graph the inequality |x - 3| ≥ 1, first, identify the critical points where the expression inside the absolute value becomes equal to ±1. Then, determine the intervals where the absolute value expression is greater than or equal to 1. Finally, graph the solution on a number line.

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Answer 3
To solve and graph the inequality \( |x - 3| \geq 1 \), follow these steps: 1. Split the inequality into two cases: a) \( x - 3 \geq 1 \) (when \( x - 3 \) is positive or zero) b) \( -(x - 3) \geq 1 \) (when \( x - 3 \) is negative) 2. Solve each case separately for \( x \): a) For \( x - 3 \geq 1 \): \( x - 3 \geq 1 \) \( x \geq 4 \) b) For \( -(x - 3) \geq 1 \): \( -(x - 3) \geq 1 \) \( -x + 3 \geq 1 \) \( -x \geq -2 \) \( x \leq 2 \) 3. Combine the solutions: The solution is \( x \leq 2 \) or \( x \geq 4 \). 4. Graph the solution on a number line: - Place an open circle at \( x = 2 \) and \( x = 4 \) to indicate that they are not included in the solution. - Shade the regions to the left of 2 and to the right of 4 to represent the solution. So, the graph of the inequality \( |x - 3| \geq 1 \) on a number line is: ``` <--|---o--------o---|---> 2 4 ``` The shaded regions represent the values of \( x \) that satisfy the inequality.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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