Match each inequality to its solution. Drag the tiles to the correct boxes. Not all tiles will be used.
Tiles
$\begin{array}{lllllll} x>3 & x<3 & x>-2 & x<-2 & x<2 & x>2 & x<-3 \end{array} \quad x>-3$
Pairs
$\begin{array}{l} 2(5)^z>250 \quad \longrightarrow \\ 4\left(\frac{1}{3}\right)^x<36 \quad \longrightarrow \\ 10\left(\frac{1}{2}\right)^x>80 \quad \longrightarrow \\ \left(\frac{1}{2}\right)(6)^x<18 \quad \longrightarrow \end{array}$
Answer (2)
The inequalities solve to the following results: 3"> z > 3 , -2"> x > − 2 , x < − 3 , and x < 2 . Each inequality is matched with its corresponding solution correctly. After solving each inequality, we have identified the correct relationships and made successful matches accordingly.
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Solve 250"> 2 ( 5 ) z > 250 and get 3"> z > 3 .
Solve 4 ( 3 1 ) x < 36 and get -2"> x > − 2 .
Solve 80"> 10 ( 2 1 ) x > 80 and get x < − 3 .
Solve ( 2 1 ) ( 6 ) x < 18 and get x < 2 .
The solutions are 3"> z > 3 , -2"> x > − 2 , x < − 3 , x < 2 , respectively. 3, x>-2, x<-3, x<2}"> z > 3 , x > − 2 , x < − 3 , x < 2
Explanation
Problem Overview We are given four exponential inequalities to solve and match with their corresponding solutions from a given set of options. Let's solve each inequality step by step.
Solving Inequality 1
Solve the inequality 250"> 2 ( 5 ) z > 250 :
First, divide both sides by 2: 125"> 5 z > 125 Since 125 = 5 3 , we have: 5^3"> 5 z > 5 3 Therefore, 3"> z > 3 .
Solving Inequality 2
Solve the inequality 4 ( 3 1 ) x < 36 :
First, divide both sides by 4: ( 3 1 ) x < 9 Since 9 = ( 3 1 ) − 2 , we have: ( 3 1 ) x < ( 3 1 ) − 2 Therefore, -2"> x > − 2 .
Solving Inequality 3
Solve the inequality 80"> 10 ( 2 1 ) x > 80 :
First, divide both sides by 10: 8"> ( 2 1 ) x > 8 Since 8 = ( 2 1 ) − 3 , we have: \left(\frac{1}{2}\right)^{-3}"> ( 2 1 ) x > ( 2 1 ) − 3 Therefore, x < − 3 .
Solving Inequality 4
Solve the inequality ( 2 1 ) ( 6 ) x < 18 :
First, multiply both sides by 2: 6 x < 36 Since 36 = 6 2 , we have: 6 x < 6 2 Therefore, x < 2 .
Matching Solutions Now, let's match each inequality with its solution:
250"> 2 ( 5 ) z > 250 corresponds to 3"> z > 3 .
4 ( 3 1 ) x < 36 corresponds to -2"> x > − 2 .
80"> 10 ( 2 1 ) x > 80 corresponds to x < − 3 .
( 2 1 ) ( 6 ) x < 18 corresponds to x < 2 .
Examples
Exponential inequalities are used in various real-world scenarios, such as modeling population growth, radioactive decay, and compound interest. For example, if you invest money in an account with compound interest, you can use exponential inequalities to determine how long it will take for your investment to reach a certain value. Similarly, in environmental science, exponential inequalities can be used to model the decay of pollutants in the environment and predict when the concentration of a pollutant will fall below a safe level. Understanding and solving exponential inequalities is crucial for making informed decisions in these and other practical applications.
