Let's solve the equation step-by-step:
1. Start with the given equation:
[tex]\[ 2(1.5)^x + 1 = -3 \][/tex]
2. First, we need to isolate the exponential term. Subtract 1 from both sides of the equation:
[tex]\[ 2(1.5)^x + 1 - 1 = -3 - 1 \][/tex]
Simplifying this, we get:
[tex]\[ 2(1.5)^x = -4 \][/tex]
3. Next, we need to isolate the base with the exponent. Divide both sides of the equation by 2:
[tex]\[ \frac{2(1.5)^x}{2} = \frac{-4}{2} \][/tex]
This simplifies to:
[tex]\[ (1.5)^x = -2 \][/tex]
4. Now we need to recall a key property of exponential functions. The expression [tex]\((1.5)^x\)[/tex] is an exponential function with a positive base (1.5). For any real number [tex]\(x\)[/tex], an exponential function with a positive base always yields a positive result. It can never be negative.
5. Since [tex]\((1.5)^x\)[/tex] is always positive for any [tex]\(x \in \mathbb{R}\)[/tex], it is impossible for it to equal -2, which is a negative number.
6. Therefore, there can be no real number [tex]\(x\)[/tex] that satisfies the equation:
[tex]\[ (1.5)^x = -2 \][/tex]
In conclusion, there are no real solutions for the equation:
[tex]\[ 2(1.5)^x + 1 = -3 \][/tex]
