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Chapter 10: Q54. (page 528)

Solve the inequality $32^{5 p + 2} 鈮� 16^{5 p}$ . Check your solution.

Short Answer

Expert verified

The solution for the inequality is $p 鈮� - 2$ .

Step by step solution

01

Step 1. Write down the given inequality.

The given inequality is $32^{5 p + 2} 鈮� 16^{5 p}$ .

02

Step 2. Simplify the inequality.

The given inequality $32^{5 p + 2} 鈮� 16^{5 p}$ is solved as:

$\begin{array}{l} 32^{5 p + 2} 鈮� 16^{5 p} .... (Given) \\ {(2^{5})}^{5 p + 2} 鈮� {(2^{4})}^{5 p} \\ {(2)}^{25 p + 10} 鈮� {(2)}^{20 p} \\ 25 p + 10 鈮� 20 p \\ 5 p 鈮� 鈭� 10 \\ p 鈮� 鈭� 2 \end{array}$

03

Step 3. Verify the result for obtained value.

Plugging $p = - 2$ in given inequality and verify the result.

$\begin{array}{l} 32^{5 p + 2} 鈮� 16^{5 p} .... (Given) \\ 32^{5 (鈭� 2) + 2} 鈮� 16^{5 (鈭� 2)} \\ 32^{鈭� 8} 鈮� 16^{鈭� 10} \\ {(2^{5})}^{鈭� 8} 鈮� {(2^{4})}^{鈭� 10} \\ {(2)}^{鈭� 40} 鈮� {(2)}^{鈭� 40} .... (True) \end{array}$

Hence the solution is verified for $p = - 2$ .

04

Step 4. Conclusion.

The solution for the given inequality is $p 鈮� - 2$ .

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