Inequalities by Example -- Section 1.7 --
 Solve for x:  3(x – 2)  £  2(x – 4)
 Solve Step 3(x – 2)  £  2(x – 4) Simplify each side first 3x – 6 £  2x – 8 Remove parenthesis 3x – 6 + 6 £  2x – 8 + 6 Add 6 to both sides 3x  £  2x – 2 Subtract 2x to both sides 3x – 2x  £  2x – 2x – 2 x  £  – 2 Inequality Notation Click Link for Inequality Grapher. (-¥, -2] Interval Notation Graph Note:  ] means we include the number. We include -2 since we have the £  symbol. Solve for x:  4(x – 3)  £  3(x – 5) [Solution]
 Solve for x:
 Solve Step Multiply all 3 parts by 3 Notice the 3's cancel 6 <  4 – 2x < 9 Now the denominator is gone! 6 – 4 <  4 – 4 – 2x < 9 – 4 Subtract 4 from each part 2 <  – 2x <  5 Look!  If we divide by a negative we have to reverse the inequality signs Divide all parts by -2 -1 > x > -5/2 Rewrite so we go from smallest to largest value -5/2 <  x < -1 Inequality Notation (-5/2, -1) Interval Notation Graph This means x has to be between -5/2 and -1 (x could not be -5/2 or -1 because of the < symbol)
 Solve for x:
 [Solution]
 Solve for x:     x2 – x  ³ 20
 I will expand upon Method 1 like example 7 on page 137. Solve Step Check x2 – x  ³ 20 First, get zero on one side.  Both methods will fail if you forget this step. (5 – 5)(5+ 4) ³ 0           0(9) = 0 ³ 0 x2 – x – 20 ³ 0 Factor. Click here for help factoring. (-4 – 5)(-4+ 4) ³ 0 (x – 5)(x+ 4) ³ 0 Clearly, 5 and -4 are solutions. -9(0) = 0 ³ 0 Since this an inequality, we expect more solutions. The possible solutions are broken up into these intervals Use -5,  0,  and 6 Pick points in these intervals to see if they work. Test Points Solution if x = -5, then (-5)2 – (-5) – 20 = 25 + 5 – 20 = 10 ³ 0  TRUE  So (–¥, –4] works.** (–¥, –4] if x = 0, then (0)2 – (0) – 20 = 0 + 0 – 20 = –20 ³ 0  FALSE   So [–5, –4] fails.** if x = 6, then (6)2 – (6) – 20 = 36 – 6 – 20 = 10 ³ 0  TRUE   So [5, ¥) works.** [5, ¥) The solutions is (–¥, –4]  È [5, ¥) **Important:  How do you know if the Test Point works?   Check the inequality symbol.  x2 – x – 20 ³ 0  If the statement is true, it works.
 Solve for x:   x2 – 2x  ³ 8
 [Solution]
 Solve for x:
 I will expand upon Method 1 like example 9 on page 138. Solve Step Since zero is already on one side, just factor the numerator. (x + 3)(x – 3) = 0 gives x =3, -3 Clearly, 3 and -3 are solutions. Check Same for -3 x + 1 = 0 gives x = -1 But!!!  Cannot divide by zero Check is undefined.  So -1 is not part of the solution. Since this an inequality, we expect more solutions. The possible solutions are broken up into these intervals Use -4, -2,  0, and 4 Pick points in these intervals to see if they work. Test Points if x = -4, then FALSE    So (–¥, –3] fails.** if x = -2, then TRUE   So [–3, –1) works.** if x = 0, then FALSE    So (–1, 3] fails.** if x = 4, then TRUE   So [3, ¥)  works.** The solutions is [–3, –1)  È [3, ¥) **Important: Remember that -1 is not included because division by zero in not allowed.
 Solve for x:
 [Solution]