Math Problem Statement
реши
Solution
Let's go through the problems one by one:
1. Solve the equation:
We can express as , so the equation becomes: Equating the exponents: Solving for :
Let’s set . Then , so the equation becomes: This is a quadratic equation. Solving it using the quadratic formula: Thus: Since must be positive, , so gives .
2. Solve the inequality:
Taking the logarithm of both sides (since , the inequality will reverse): Since is negative, we divide by it and reverse the inequality: Calculating the result gives the value for .
3. Solve the system of equations:
x - y = 4 \\ 5x + y = 25 \end{cases}$$ Adding the two equations to eliminate $$y$$: $$(x - y) + (5x + y) = 4 + 25 \quad \Rightarrow \quad 6x = 29 \quad \Rightarrow \quad x = \frac{29}{6}$$ Substituting $$x = \frac{29}{6}$$ into the first equation: $$\frac{29}{6} - y = 4 \quad \Rightarrow \quad y = \frac{29}{6} - 4 = \frac{29}{6} - \frac{24}{6} = \frac{5}{6}$$ Thus, $$x = \frac{29}{6}$$ and $$y = \frac{5}{6}$$. ### 4. Solve the inequality: 1) $$\left(\sqrt{5}\right)^{x - 6} < \frac{1}{5}$$ Express $$\frac{1}{5}$$ as $$5^{-1}$$, and $$\sqrt{5}$$ as $$5^{1/2}$$: $$\left(5^{1/2}\right)^{x - 6} < 5^{-1}$$ Simplifying the exponent: $$5^{\frac{x - 6}{2}} < 5^{-1}$$ Equating the exponents: $$\frac{x - 6}{2} < -1 \quad \Rightarrow \quad x - 6 < -2 \quad \Rightarrow \quad x < 4$$ 2) $$\left(\frac{2}{13}\right)^{x^2 - 1} \geq 1$$ Since any base raised to 0 equals 1, we solve: $$x^2 - 1 \geq 0 \quad \Rightarrow \quad x^2 \geq 1 \quad \Rightarrow \quad x \geq 1 \quad \text{or} \quad x \leq -1$$ ### 5. Solve the equation: $$7^{x+1} + 3 \cdot 7^x = 2x^5 + 3 \cdot 2x$$ This equation seems to involve complex exponential and polynomial terms, and would require further methods like substitution or numerical solving depending on the given terms. --- Let me know if you'd like more details on any specific step, or if you have any questions! Here are 5 related problems: 1. Solve the equation $$\left(\frac{1}{4}\right)^{x-2} = 16$$. 2. Solve the inequality $$\left(\frac{5}{6}\right)^x < \frac{2}{5}$$. 3. Solve the system of equations $$\begin{cases} 2x + 3y = 7 \\ 4x - y = 5 \end{cases}$$. 4. Solve the inequality $$\left(2^{x-3}\right) > \frac{1}{8}$$. 5. Solve the equation $$5^{x+2} + 25^{x-1} = 30$$. **Tip:** When solving systems of equations, elimination or substitution are both powerful methods! Choose the one that simplifies the problem faster.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Exponential equations
Quadratic equations
Inequalities
System of linear equations
Formulas
Exponential form transformation
Quadratic formula: ax^2 + bx + c = 0
Logarithmic properties
Linear equation solving methods
Theorems
Properties of exponents
Quadratic equation solutions
Logarithm rules
Suitable Grade Level
Grades 9-11