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College Algebra K/DCWednesday, 03 December 2014
• OBJECTIVE TSW solve and graph the solutions to linear inequalities, three-part inequalities, quadratic inequalities, and rational inequalities.
• ASSIGNMENT DUE– Sec. 1.6: p. 143 (7-12 all, 15-25 odd, 35-42 all, 44-
52 even) wire basket
1-2
Inequalities1.7Linear Inequalities ▪ Three-Part Inequalities
1-3
Subtract 7.
Divide by –2. Reverse the direction of the inequality symbol when multiplying or dividing by a negative number ! !
Solution set:
Solving a Linear Inequality
interval nota on6, ti
set-builder not i6 t n| a ox x
Havens’ Preference
Use interval notation.
Solve & graph:
1-4
Solving a Linear Inequality
Subtract 8.
Add 4x.
Divide by 6.
Solution set:Write the solution set in interval notation and graph it.
5 4 2x x 45 44 2xx x x
Solve & graph:
1-5
Using Brackets vs. Parentheses
With a strict inequality: Use parentheses.
Ex: x > 4
½ > x
With an inequality that includes equals, use a bracket.
Ex: x ≤ 9
x ≥ –1
“Infinity” and “Negative Infinity” ALWAYS get parentheses!!!
4,
1, 2
, 9 1,
1-6
Interval Notation
Always write the ends in numerical order.
Ex: x < –9 (–∞, –9)
not (–9, –∞)
Ex: 8 ≤ x [8, ∞)
not (∞, 8]
Make sure your brackets look like brackets and not squared-off parentheses!
Make sure your parentheses look like parentheses and not rounded-off brackets!
BOTTOM LINE: Write your answers neatly!
1-7
Solving a Three-Part Inequality
Add 8 to all three parts.
Divide by 6.
Write the solution set in interval notation and graph it.
Solution set:
Solve & graph:
1-8
Assignment: Sec. 1.7: p. 155 (13-33 odd) Due before you go to lunch today (black tray).
Solve each inequality. Write each solution set in interval notation and graph.
3) 2 11 8 6x 25) 2 11 x x
2 5 1 5 317) x x 8 3 29 2 71 ) x x x
4 721) 2 5
3
xx
1 2 1 13
3 5 2 1023) x x x
5 5 2 125) 1x 1027) 1 2 4 6x
11 3 1 1729) x 1
431) 52
x
4333) 4
5
x
1-9
Inequalities1.7Quadratic Inequalities ▪ Rational Inequalities
1-10
Solving a Quadratic Inequality
Solve .
Step 1: Find the values of x that satisfy .
or
or
Step 2: The two numbers divide a number line into three regions.
Use closed dots since the inequality symbol includes equality.
1-11
Solving a Quadratic Inequality
Solve .
Step 3: Choose any value in each of the intervals and find the sign of in each of the intervals. 5 3x x
–4 60
4 4 4: 5 3 9 1x
+
0 0: 5 3 5 30x
6 6 6: 5 3 1 9x
– +
1-12
Solving a Quadratic Inequality
Solve .
–4 60+ – +
Step 4: Since (negative), the solution is the interval that makes negative.
2 2 15 0x x 5 3x x
Solution set: [–3, 5]
23 10 7x x
1-13
Solving Quadratic Inequalities
Solve and graph each of the following:
a)
b)
22 3 0x x
1-14
Solving Rational Inequalities
Solve and graph each of the following:
d)
e)
f)
50
3xx
20
1xx
63
2x
x
Assignment: Sec. 1.7: pp. 155-156 (39-44 all, 69-71 all, 73-77 odd, 78) Due on Wednesday, 10 December 2014 (TEST day).
Solve each inequality. Write each solution in interval notation.2 0) 639 x x 2 7 1040 0) x x
2 42) 34 x x
241 2 9 1) 8x x
2 6 16 844) x x 2 ) 4 6 343 x x
00) 4
71x
x
09) 5
63x
x 2
1)1
17x
x
36
) 73 2x
47 5
15)
x
10 5
377)
2x
1
7 3)2
8x