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Definition of the exponential inequalities An inequality with variables in the exponent is called an Exponential Inequality.
For example
2x<23 3x>81 5−x≥25
Method for solving exponential inequalities
(Because)(1)When a>1 ax>a p → x>p (2)When 0<a<1 ax>a p → x<p (1)When a>1, as shown in Figure 1, the graph of y=ax is increasing. ax>a p ⇔ x>p (2)When 0<a<1, as shown in Figure 2, the graph of y=ax is decreasing. ax>a p ⇔ x<p
Caution!
In case (2), the direction of the inequality symbol must be reversed. |
Figure 1
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(1)When a>1
ax<a p → x<p ax≥a p → x≥ p ax≤a p → x≤ p (2)When 0<a<1 ax<a p → x>p ax≥a p → x≤ p ax≤a p → x≥ p |
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ExamplesSolve the following inequalities for x.
(1)2x>32
(Answer)Rewrite the right side with base 2. 2x>25 x>5
(2)22x−4≤ 22−x
(Answer)As the bases are the same and greater than 1. 2x−4≤ 2−x Solve this linear inequality for x. 3x≤ 6 x≤ 2 |
(3)0.5x−1< 0.252
(Answer)Rewrite the right side with base 0.5. 0.5x−1< (0.52)2 0.5x−1< 0.54 As the bases are the same and less than 1. x−1>4 x>5 |
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Question 1Solve the following inequalities for x. (Choose the correct answer. Maybe these are difficult only by mental arithmetic, you may use calculation sheet.) x>−2 x≥−2 x<−2 x≤−2 x>3 x≥3 x<3 x≤3 0<x<3 0<x≤3 |
Rewrite the right side using base 9.
9x+1>92 As the bases are greater than 1. x+1>2 x>1 |
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x>−2 x≥−2 x<−2 x≤−2 x>3 x≥3 x<3 x≤3 0<x<3 0<x≤3 |
Rewrite both sides using base 2.
(22)x+3≤(23)x+1 22x+6≤23x+3 As the bases are greater than 1. 2x+6≤3x+3 −x≤−3 x≥3 |
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x>−2 x≥−2 x<−2 x≤−2 x>3 x≥3 x<3 x≤3 0<x<3 0<x≤3 |
Rewrite the right side using base .
( )2x+1<(( )3)x ( )2x+1<( )3x As the bases are less than 1. 2x+1>3x −x>−1 x<1
As shown in Figure 2, y=ax is defined for −∞<x<∞, the condition x>0 is not needed.
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x>−2 x≥−2 x<−2 x≤−2 x>3 x≥3 x<3 x≤3 0<x<3 0<x≤3 |
Rewrite both sides using base 5.
(5)x≥ 5−1 5≥ 5−1 As the bases are greater than 1. ≥−1 x≥−2 |
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x>−2 x≥−2 x<−2 x≤−2 x>3 x≥3 x<3 x≤3 0<x<3 0<x≤3 |
Rewrite both sides using base 10.
(10)1−x>(10−1)x+1 10(1−x)>10−x−1 As the bases are greater than 1. (1−x)>−x−1 1−x>−3x−3 2x>−4 x>−2 |
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x>−2 x≥−2 x<−2 x≤−2 x>3 x≥3 x<3 x≤3 0<x<3 0<x≤3 |
Rewrite both sides using base .
( )2x+1≤(( )3)2−x ( )2x+1≤( )6−3x As the bases are less than 1. 2x+1≥6−3x 5x≥5 x≥1 |
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Method for solving exponential inequalities
(Notice)Suppose a>0, a≠1 When ax appears repeatedly, let ax=X and solve for X.
As shown in Figure 1 and Figure 2, when ax=X
•x can be real value. (−∞<x<∞) •X should be only positive value. (X>0) As 4x=(22)x=(2x)2, introducing a new variable 2x=X, 4x is expressed in the form X 2. As 9x=(32)x=(3x)2, introducing a new variable 3x=X, 9x is expressed in the form X 2. ExamplesSolve the following equations for x.
1.4x−2x−12<0
(Answer)Express by 2x (22)x−2x−12<0 22x−2x−12<0 (2x)2−2x−12<0 Let 2x=X(>0) …(1) X 2−X−12<0 Factorize the left side. (X+3)(X−4)<0 −3<X<4 …(2) (1)(2)→ 0<X<4 X=2x>0 is satisfied. X=2x<4=22 → x<2 Therefore x<2
2.3×9x+2×3x−1>0
(Answer)Express by 3x 3×(32)x+2×3x−1>0 3×32x+2×3x−1>0 3×(3x)2+2×3x−1>0 Let 3x=X(>0) …(1) 3X 2+2X−1>0 Factorize the left side. (X+1)(3X−1)>0 X<−1 or X> …(2) (1)(2)→ X> X=3x>=3−1 x>−1 |
Figure 3
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Question 2Solve the following inequalities for x. (Choose the correct answer. Maybe these are difficult only by mental arithmetic, you may use calculation sheet.) x<−1 x>−1 x<0, x>1 x<1, x>2 |
Rewrite using base 2.
3×2x−(22)x<2 3×2x−(2x)2<2 Let 2x=X(>0) …(A) 3X−X 2<2 X 2−3X+2>0 (X−1)(X−2)>0 X<1 or X>2 …(B) (A)(B)→ 0<X<1 or X>2 0<X<1 → (0<)2x<20 → x<0 X>2 → 2x>21 → x>1 Thus, x<0,x>1 |
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x<−1 x>−1 x<0, x>1 x<1, x>2 |
Rewrite using base 2.
As 9x=(3x)2 Let 3x=X(>0) …(A) X 2−4X+3<0 Factorize the left side. (X−1)(X−3)<0 1<X<3 …(B) (A)(B)→ 1<X<3 1<3x<3 → 30<3x<31 → 0<x<1 |
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x<−1 x>−1 x<0, x>1 x<1, x>2 |
Rewrite the left side using base 0.5.
As 0.25x=(0.5x)2 Let 0.5x=X(>0) …(A) X 2−X−2>0 Factorize the left side. (X+1)(X−2)>0 X<−1 or X>2…(B) (A)(B)→ X>2 0.5x>2 → 0.5x>( )−1=0.5−1 → x<−1 |