Select the correct answer. What is the solution to this inequality? [tex]$3\left(\frac{1}{4}\right)^{x+1}\ \textless \ 192$[/tex]
A. [tex]$x\ \textgreater \ -4$[/tex]
B. [tex]$x\ \textless \ 6$[/tex]
C. [tex]$x\ \textless \ 4$[/tex]
D. [tex]$x\ \textgreater \ -6$[/tex]
Answer (2)
Divide both sides by 3: ( 4 1 ) x + 1 < 64 .
Rewrite with the same base: 4 − ( x + 1 ) < 4 3 .
Compare exponents: − ( x + 1 ) < 3 .
Solve for x : -4"> x > − 4 .
Explanation
Understanding the Inequality We are given the inequality 3 ( 4 1 ) x + 1 < 192 . Our goal is to isolate x and determine the solution set.
Dividing by 3 First, divide both sides of the inequality by 3: ( 4 1 ) x + 1 < 3 192
Simplifying the Right Side Simplify the right side: ( 4 1 ) x + 1 < 64
Rewriting with the Same Base Rewrite both sides with the same base. Since 4 1 = 4 − 1 and 64 = 4 3 , we have ( 4 − 1 ) x + 1 < 4 3
Simplifying the Exponent Simplify the left side using the power of a power rule: 4 − ( x + 1 ) < 4 3
Comparing Exponents Since the base is greater than 1, we can compare the exponents directly: − ( x + 1 ) < 3
Distributing the Negative Sign Distribute the negative sign: − x − 1 < 3
Adding 1 to Both Sides Add 1 to both sides: − x < 4
Multiplying by -1 Multiply both sides by -1 and flip the inequality sign: -4"> x > − 4
Examples
Understanding inequalities is crucial in various real-world scenarios. For example, when planning a budget, you might use inequalities to determine how much you can spend on different items while staying within your income. Similarly, in science, inequalities can help define the range of acceptable values for experimental parameters to ensure accurate results. In business, inequalities are used to optimize production and maximize profit within given constraints.
The solution to the inequality 3 ( 4 1 ) x + 1 < 192 is -4"> x > − 4 , making the correct answer option A. This was found by isolating the exponential term and comparing exponents after converting bases.
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