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How do you solve #|x - 7| <10#?

Answer 1

See the entire solution process below:

To solve an absolute value inequality you must solve for both the negative and positive form of what the term within the absolute value function is compared to using a complex inequality.

#abs(x - 7) < 10->#

#-10 < x - 7 < 10#

Now, add #color(red)(7)# to each section of the complex inequality to solve for #x# while keeping the inequality balanced:

#-10 + color(red)(7) < x - 7 + color(red)(7) < 10 + color(red)(7)#

#-3 < x - 0 < 17#

#-3 < x < 17#

Or, in interval form:

(-3, 17)

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Answer 2

Genesis Allen

To solve the inequality |x - 7| < 10, you need to consider two cases:

When ( x - 7 ) is positive: ( x - 7 < 10 )
When ( x - 7 ) is negative: ( -(x - 7) < 10 )

For the first case: ( x - 7 < 10 ) Add 7 to both sides: ( x < 17 )

For the second case: ( -(x - 7) < 10 ) Distribute the negative sign: ( -x + 7 < 10 ) Subtract 7 from both sides: ( -x < 3 ) Multiply both sides by -1 (remembering to reverse the inequality sign): ( x > -3 )

So, the solution is ( -3 < x < 17 ).

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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.

How do you solve #|x - 7| &lt;10#?

How do you solve #|x - 7| <10#?