- Isolate the exponential expression by subtracting 5 and multiplying by -1, which flips the inequality sign to get \(3^x < 9\).
- Convert 9 to the base 3 power: \(3^x < 3^2\).
- Since the base is greater than 1, compare the exponents to get \(x < 2\).
- The solution set is \((-\infty, 2)\), which corresponds to the first option: \(\boxed{(-\infty, 2)}\).
Explanation
- Analyzing the Inequality
We start by analyzing the given inequality: \(-3^x + 5 > -4\). Our goal is to determine the set of values for \(x\) that satisfy this condition. The inequality involves an exponential function, which requires careful handling of algebraic operations. To solve for \(x\), we first aim to isolate the exponential expression \(3^x\).
- Isolating the Exponential Term
Starting with \(-3^x + 5 > -4\), we subtract 5 from both sides of the inequality:
\[ -3^x > -4 - 5 \]
\[ -3^x > -9 \]
Now, we multiply both sides of the inequality by \(-1\). Remember that multiplying or dividing an inequality by a negative number reverses the direction of the inequality sign:
\[ 3^x < 9 \]
- Solving for x
Next, we express both sides of the inequality in terms of the same base. Since \(9 = 3^2\), we can rewrite the inequality as:
\[ 3^x < 3^2 \]
Because the base of the exponential function \(f(x) = 3^x\) is greater than 1, the function is strictly increasing. This means that if \(3^a < 3^b\), then \(a < b\). Therefore, we can conclude:
\[ x < 2 \]
- Conclusion and Selection
The inequality \(x < 2\) means that \(x\) can be any real number less than 2. In interval notation, this set of values is expressed as \((-\infty, 2)\). Comparing this to the given choices, we see that it corresponds to the first option.
Examples
In real life, inequalities like this appear when modeling exponential growth or decay. For instance, if you are tracking the temperature of a cooling object or the growth of a population that is limited by a certain threshold, solving an inequality allows you to determine the timeframe during which a specific condition—such as "temperature stays above a certain level"—is met. Understanding how to solve these equations helps in predicting when a system will reach equilibrium or fall below a critical value.