Download 4√ 81 4√ 81

NAME
DATE
7-2
PERIOD
Study Guide and Intervention
Solving Exponential Equations and Inequalities
Solve Exponential Equations All the properties of rational exponents that you know
also apply to real exponents. Remember that am · an = am + n, (am)n = amn, and am ÷ an = am - n.
Property of Equality for
Exponential Functions
If b is a positive number other than 1, then bx = by if and only if x = y.
Example 2
Write an exponential
function whose graph passes through the
points (0, 3) and (4, 81).
Example 1
Solve 4 x - 1 = 2 x + 5.
4x - 1 = 2x + 5
Original equation
2 x-1
(2 )
= 2x + 5
Rewrite 4 as 22.
2(x - 1) = x + 5 Prop. of Inequality for Exponential
Distributive Property
The y-intercept is (0, 3), so a = 3. Since the
4 81
−.
other point is (4, 81), b = Subtract x and add 2 to
each side.
Simplifying
Functions
2x - 2 = x + 5
x=7
√3
81
−
= √
27 ≈ 2.280, the equation
√
3
4
4
is y = 3(2.280)x.
Exercises
Solve each equation.
3
4. 4 x + 1 = 8 2x + 3
7
-−
4
7. 9 x + 1 = 27 x + 4
-10
1
-−
2. 23x = 4x + 2
4
3. 3 2x - 1 = −19
1
5. 8x - 2 = −
16
2
−
6. 25 2x = 125 x + 2 6
3
8. 362x + 4 = 216
x+5
( 64 )
1
9. −
x-2
2
= 16
3x + 1
4
−
7
9
Write an exponential function for the graph that passes through the given points.
10. (0, 4) and (2, 36)
y = 4(3)x
13. (0, 2) and (5, 486)
x
y = 2(3)
16. (0, 3) and (3, 24)
y = 3(2)x
Chapter 7
11. (0, 6) and (1, 81)
y = 6(13.5)x
(
y = 5(2)x
27
14. (0, 8) and 3, −
()
3
y=8 −
4
12. (0, 5) and (6, 320)
8
)
15. (0, 1) and (4, 625)
x
y = (5)x
17. (0, 12) and (4, 144)
18. (0, 9) and (2, 49)
y = 12(1.861)x
y = 9(2.333)x
12
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. 3 2x - 1 = 3 x + 2
NAME
DATE
7-2
Study Guide and Intervention
PERIOD
(continued)
Solving Exponential Equations and Inequalities
Solve Exponential Inequalities
An exponential inequality is an inequality
involving exponential functions.
Property of Inequality for
Exponential Functions
1
52x - 1 > −
125
2x - 1
5
> 5-3
2x - 1 > -3
2x > -2
x > -1
1
Solve 52x - 1 > −
.
125
Original inequality
1
Rewrite −
as 5-3.
125
Prop. of Inequality for Exponential Functions
Add 1 to each side.
Divide each side by 2.
Lesson 7-2
Example
If b > 1
then bx > by if and only if x > y
and bx < by if and only if x < y.
The solution set is {x | x > -1}.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each inequality.
1
1. 3 x - 4 < −
27
x<1
3. 5 2x < 125 x - 5
5
x>−
x > 15
3
4. 10 4x + 1 > 100 x - 2
5
x > -−
2
7. 16 ≥ 4x + 5
x ≤ -3
2x - 4
1
10. −
<9
81
5. 7 3x < 49 1 - x
6. 8 2x - 5 < 4 x + 8
31
x<−
2
x<−
5
4
( 27 )
1
8. −
2x + 1
( 243 )
1
≤ −
3x - 2
13
x≤−
( 25
11. 323x - 4 > 1284x + 3
1
x ≥ -−
13
x-3
( 343 )
7
14. −
x-3
> 82 x
7
12. 27
2x - 5
1 5x
< −
)
(9
15
x<−
16
13
3x + 1
1 2x - 1
13. −
≤ 125
)
(2)
1
9. −
3
x<−
9
41
x < -−
x>1
Chapter 7
2. 42x - 2 > 2 x + 1
( 49
2x + 1
1
≥ −
)
x ≥ -4
( 27
6x - 1
9
15. −
)
27 -x + 6
≥ −
)
(9
x ≤ -1
13
Glencoe Algebra 2