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hello welcome to we learn
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daily in this video we're going to solve
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the equation where the absolute value of
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the difference between X and 2 equals
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the absolute value of the sum of X and 5
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as usual before watching try to solve
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this equation on your own let's start
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we'll solve this equation using two
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methods the first method is based on the
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geometric meaning of absolute value
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specifically we need to remember that
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the absolute value of the difference
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between two numbers A and B represents
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the distance between points A and B on
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the number line regardless of which
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number is larger to use the geometric
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meaning of absolute value we need to
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express each absolute value expression
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as a difference this means we leave the
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first absolute value as it is the
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absolute value of x - 2 and rewrite the
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second absolute value the absolute value
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of x + 5 as the absolute value of x -
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-5 now the left side represents the
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distance from point x to 2 and the right
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side represents the distance from point
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x to -5 these distances must be equal
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which means the point x must be
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equidistant from -5 and 2 this condition
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can only be true if the point x
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coincides with the midpoint of the
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2 so we can find the value of x by
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calculating the arithmetic mean of of -5
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2 adding -5 and 2 then dividing by two
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gives us -3 halves or in decimal form -
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1 and a half now for the second
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method since absolute value is always
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non- negative we conclude that both the
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left and right sides of the equation are
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greater than or equal to zero this
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allows us to square both sides of the
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equation without changing the solution
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so we can replace the the original
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equation with the equality of the
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squares of these absolute values this
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means the square of the first absolute
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value x - 2 must equal the square of the
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second absolute value x + 5 we recall
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that the square of an absolute value is
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equal to the square of the expression
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inside the absolute value this allows us
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to replace the absolute value signs with
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regular parentheses giving us a simpler
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equation the square of x - 2 = x + 5
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next we move everything to one side of
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the equation resulting in the difference
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of squares the square of x - 2 - the
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0 using the difference of squares
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formula the left side can be factored
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into x - 2 - x - 5 multiplied x - 2 + x
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+ 5 simplifying the Expressions inside
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the parentheses the first Factor becomes
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-7 and the second Factor becomes 2 * x +
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3 setting the product equal to 0 we
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divide both sides by -7 leaving us with
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2 * x + 3 = 0 solving for x we get 2 * x
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= -3 which simplifies to x = -3 or - 1
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and A2 finally we write the answer x =
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-1 A2 the problem is solved
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if you found the solution helpful please
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like the video subscribe and leave a
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comment thank you for watching goodbye
