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How do you solve the inequality #|3x-2| + 4<= 7#?

Answer 1

Avery Alexander

The solution is #x in [-1/3,5/3]#

The disparity is

#|3x-2|+4<=7#

#|3x-2|<=7-4\=3#

Consequently,

#{(-3x\=3),(-3x+2<=3):}#

#~(3x<=5), #{(3x>=-1):}#

#~(x<=5/3),(x>=-1/3):}#, #<=>#

The resolution is

#x in [-1/3,5/3]#

graph{|3x-2|-3 [-2.772, 2.777, 5.55, 5.55]}

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Answer 2

Avery Alexander

To solve the inequality ( |3x - 2| + 4 \leq 7 ):

Subtract 4 from both sides: [ |3x - 2| \leq 3 ]
Solve for both the positive and negative cases: For ( 3x - 2 \geq 0 ): [ 3x - 2 \leq 3 ] For ( 3x - 2 < 0 ): [ -(3x - 2) \leq 3 ]
Solve each case separately: Case 1: ( 3x - 2 \geq 0 ) [ 3x - 2 \leq 3 ] [ 3x \leq 5 ] [ x \leq \frac{5}{3} ]

Case 2: ( 3x - 2 < 0 ) [ -(3x - 2) \leq 3 ] [ -3x + 2 \leq 3 ] [ -3x \leq 1 ] [ x \geq -\frac{1}{3} ]
Combine the solutions: [ -\frac{1}{3} \leq x \leq \frac{5}{3} ]

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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 the inequality #|3x-2| + 4&lt;= 7#?

How do you solve the inequality #|3x-2| + 4<= 7#?