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solve the absolute value inequality. |2(x-1)+8| less than or equal to 6.
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[-6 , 0]

Step-by-step explanation:

If the absolute value of an expression is equal to a number, that means that the expression itself could be equal to either the negative equivalent to that number or the positive equivalent.

for example, if my inequality is |x| > -3

x could either be:

-3 or 3

So,

| 2(x - 1) + 8 | ≤ 6

can be separated into two separate inequalities:

2(x - 1) + 8 ≤ 6

or, 2(x - 1) + 8 ≥ - 6

we solve these inequalities separately.

2(x - 1) + 8 ≤ 6

2x - 2 + 8 ≤ 6 [distribute 2]

2x ≤ 0 [add 2 to both sides, subtract 8 from both sides]

x ≤ 0 [finalize isolating x by dividing both sides of the equation by 2]

expressed as [in interval notation]:

(-∞, 0]

now, let's solve for the other inequality.

2(x - 1) + 8 ≥ - 6

2x - 2 + 8 ≥ - 6 [distribute 2]

2x ≥ -12 [add 2 to both sides, subtract 8 from both]

x ≥ -6 [divide both by 2 to isolate x]

expressed as [in interval notation]:

[-6 , ∞)

So, our answer for x is going to be

-6 ≤ x ≤ 0

or, expressed in interval notation:

[-6 , 0]

*[ includes number]

*( is not equal to number)
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