Answers:
The absolute value of a real number (also called modulus) is a "non-negative value of that number without regard to its sign". This is because absolute values are, in fact, distances.
In other words: An absolute value is a number's distance from zero in the Number line, and are solved in the following way:
a) [tex]|3x-5|=4[/tex]
[tex]3x-5=4[/tex]
[tex]3x=4+5[/tex]
[tex]x=3[/tex]
or
[tex]3x-5=-4[/tex]
[tex]3x=-4+5[/tex]
[tex]x=\frac{1}{3}[/tex]
The value of [tex]x[/tex] is between [tex]{\frac{1}{3},3}[/tex]
b) [tex]|5x-3|=\frac{2}{3}[/tex]
[tex]5x-3=\frac{2}{3}[/tex]
[tex]5x=\frac{2}{3}+3[/tex]
[tex]x=\frac{11}{15}[/tex]
or
[tex]5x-3=-\frac{2}{3}[/tex]
[tex]5x=-\frac{2}{3}+3[/tex]
[tex]x=\frac{7}{15}[/tex]
The value of [tex]x[/tex] is between [tex]{\frac{7}{15},\frac{11}{15}}[/tex]
c) [tex]|\frac{3x-2}{2}+5|=10[/tex]
[tex]\frac{3x-2}{2}+5=10[/tex]
[tex]3x-2=10[/tex]
[tex]x=4[/tex]
or
[tex]\frac{3x-2}{2}+5=-10[/tex]
[tex]3x-2=-30[/tex]
[tex]x=-\frac{28}{3}[/tex]
The value of [tex]x[/tex] is between [tex]{-\frac{28}{3},4}[/tex]
d) [tex]|\frac{x+3}{4}-\frac{1}{2}|=3[/tex]
[tex]|\frac{x+3}{4}-\frac{1}{2}|=3[/tex]
[tex]\frac{x+3}{4}-\frac{1}{2}=3[/tex]
[tex]\frac{x+3}{4}=3+\frac{1}{2}[/tex]
[tex]x+3=14[/tex]
[tex]x=11[/tex]
or
[tex]\frac{x+3}{4}-\frac{1}{2}=-3[/tex]
[tex]\frac{x+3}{4}=-3+\frac{1}{2}[/tex]
[tex]x+3=-10[/tex]
[tex]x=-13[/tex]
The value of [tex]x[/tex] is between [tex]{-13,11}[/tex]
