Solving Linear Inequalities

I. Notations:

You should know three type of notations (1) inequality notations, (2) real number line notations and (3) interval notations. (In the test, when solving inequalities, be sure to use interval notations as your answers.)

II. Solving Inequalities involving ''AND'' and ''OR''. Note that ''AND'' means taking ''Intersections'', usually, means smaller of two sets. On the other hand, ''OR'' means taking union, and usually means taking larger of two sets.

  1. tex2html_wrap_inline18 [This means -3<2x+5 and tex2html_wrap_inline22 So you can treat this inequality as two separate inequalities and take the ''intersection'' later on. Alternatively, notice that the we can substract 5 from both ends of the inequalities, yields, tex2html_wrap_inline26 Divide 2 both sides, we get tex2html_wrap_inline30 Therefore, the answer is (-4,1].]
  2. tex2html_wrap_inline34 or tex2html_wrap_inline36 [First, tex2html_wrap_inline38 so, tex2html_wrap_inline40 Next 3x >= 9 yields, x>=3. By taking the union of these two inequalities, we get (-infinity, -2] union [3, infinity].
  3. tex2html_wrap_inline34 and tex2html_wrap_inline36 [This is to take the intersection of the previous problem, so the answer should be no solution.]
  4. 7-3x>x+4 or 3x+4<3. [First, we get 3>4x, which means tex2html_wrap_inline58 Next 3x<-1, or tex2html _wrap_inline62 Now take the union of these two inequalities, we get tex2html_wrap_inline64 ]
  5. 7-3x>x+4 and 3x+4<3 [By taking intersection of these two inequalities, yields, tex2html_wrap_inline70

Type III. Solving absolute value problems

  1. tex2html_wrap_inline72 For this type of problem, we rewrite the inequality into an ''intersection'' type of inequality: -1<3x-1<1, which yields, tex2html_wrap_inline76 so the answer is tex2html_wrap_inline78
  2. tex2html_wrap_inline80 For this type of problem, we rewrite it into a ''union'' problem: 3x-1>1 or 3x-1<-1, which means tex2html_wrap_inline86 or x<0. So the solution is tex2html_wrap_inline90
  3. tex2html_wrap_inline92 [ We can multiply both sides by 2 first, and the inequality becomes tex2html_wrap_inline96, so the solution is tex2html_wrap_inline98
  4. tex2html_wrap_inline100 [Notice that since tex2html_wrap_inline102 the inequality is the same as tex2html_wrap_inline104 so the solution is the same as the previous problem.]