Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Algebra - Absolute values.
Strategies for solving equations.

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When handling absolute value equations, my advice to you is as follows:

Only one absolute term.	1. rewrite the equation by first replacing the absolute value symbols | | with brackets and then write an implicit or explicit + ve sign in front. Solve that equation in the usual way.	Solve |2x + 3| = 11 (2x + 3) = 11 2x = 8 x = 4
	2. rewrite the equation again by replacing the absolute value symbols with brackets but this time using an explicit - ve sign in front. Be careful of double negative signs when expanding.	-(2x + 3) = 11 -2x - 3 = 11 -2x = 14 x = -7
	NOTE: do not put the + and - signs in front of other terms. It is always the term in the absolute value symbols | | which should change in sign - by definition.
	If there are other terms in x and/or numbers, just write them again as they appear. That approach also includes situations where there are numerical fractions - here just multiply through and get rid of the denominators.
		|2x - 1| = x + 2 (2x - 1) = x + 2 x = 3
		-(2x - 1) = x + 2 -2x + 1 = x + 2 -3x = 1 x = -1/3
Two absolute value terms.	Replace the absolute value symbols | | for both terms with brackets. Choose one set of terms which had the | | symbols previously. Then:	|3x - 1| = |2x + 5|   (3x - 1) = (2x + 5)
	first: write a + ve sign in front of those brackets and solve the equation;	+(3x - 1) = (2x + 5) x = 6
	second: write a - ve sign in front of those same brackets. Expand the brackets carefully (watching for negative problems) and solve the equation.	-(3x - 1) = (2x + 5) -3x + 1 = 2x + 5 -5x = 4 x = -4/5 = -0.8
Equations with absolute value fractions.	There are two approaches possible. The easiest is to rewrite the fraction as two distinct absolute value expressions. As both are positive, multiplying through to clear the denominator raises no issues.
	Then proceed as we did above for the two absolute value terms.
	Always check your answers by substituting into the original equation - here all was good!! Sometimes one solution does not work.