Statements Without Absolute Values

[nycmath]

Precalculus
David Cohen
Edition 3
Chapter 1, Section 1.1

Rewrite each expression in a form that does not contain Absolute values.

28. | x - 3 |, where x < 3

34. | x - 3 | + | x - 4 |, where x > 4

[Math Tutor]

Check the sign of what is inside the bars, then drop them accordingly.

28. If [tex]x<3[/tex] then [tex]x-3<0[/tex], so the bars flip the sign:

[tex]|x-3|=-(x-3)=3-x[/tex]

(Check: [tex]x=1[/tex] gives [tex]|-2|=2[/tex] and [tex]3-1=2[/tex].)

34. If [tex]x>4[/tex] then [tex]x-3>0[/tex] and [tex]x-4>0[/tex], so both sets of bars come off as they are:

[tex]|x-3|+|x-4|=(x-3)+(x-4)=2x-7[/tex]

(Check: [tex]x=10[/tex] gives [tex]7+6=13=2(10)-7[/tex].)

[nycmath]

Check the sign of what is inside the bars, then drop them accordingly.

28. If [tex]x<3[/tex] then [tex]x-3<0[/tex], so the bars flip the sign:

[tex]|x-3|=-(x-3)=3-x[/tex]

(Check: [tex]x=1[/tex] gives [tex]|-2|=2[/tex] and [tex]3-1=2[/tex].)

34. If [tex]x>4[/tex] then [tex]x-3>0[/tex] and [tex]x-4>0[/tex], so both sets of bars come off as they are:

[tex]|x-3|+|x-4|=(x-3)+(x-4)=2x-7[/tex]

(Check: [tex]x=10[/tex] gives [tex]7+6=13=2(10)-7[/tex].)

Is there a general absolute value rule(s) that applies here?