graph a system of equations to solve $\log (-5.6x + 1.3) = -1 - x$. round to the nearest tenth.

from the least to the greatest, the solutions are:

$x \approx$ and $x \approx 0.2$

Question

graph a system of equations to solve $\log (-5.6x + 1.3) = -1 - x$. round to the nearest tenth.

from the least to the greatest, the solutions are:

$x \approx$   and $x \approx 0.2$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Solving Exponential Equations Graphically", "Transcendental Equation Solving" ], "new_concepts": [], "current_concepts": [ "Solving Exponential Equations Graphically", "Transcendental Equation Solving", "Logarithmic Functions" ] } ## Define the system of equations Using the *Solving Exponential Equations Graphically* and *Transcendental Equation Solving* knowledge points $$ \begin{aligned} y_1 &= \log_{10}(-5.6x + 1.3) \ y_2 &= -1 - x \end{aligned} $$ ## Determine the domain of the logarithmic function Using the *Logarithmic Functions* knowledge point $$ \begin{aligned} -5.6x + 1.3 &> 0 \ -5.6x &> -1.3 \ x &< \frac{1.3}{5.6} \approx 0.232 \end{aligned} $$ ## Find the intersection points numerically Using the *Transcendental Equation Solving* knowledge point $$ \begin{aligned} &\text{Let } f(x) = \log_{10}(-5.6x + 1.3) + x + 1 = 0 \ &f(0.2) = \log_{10}(-5.6(0.2) + 1.3) + 0.2 + 1 = \log_{10}(0.1) + 1.2 = -1 + 1.2 = 0.2 \approx 0 \ &f(-1.5) = \log_{10}(-5.6(-1.5) + 1.3) - 1.5 + 1 = \log_{10}(9.7) - 0.5 \approx 0.987 - 0.5 = 0.487 \ &f(-2.0) = \log_{10}(-5.6(-2.0) + 1.3) - 2.0 + 1 = \log_{10}(12.5) - 1.0 \approx 1.097 - 1.0 = 0.097 \ &f(-2.1) = \log_{10}(-5.6(-2.1) + 1.3) - 2.1 + 1 = \log_{10}(13.06) - 1.1 \approx 1.116 - 1.1 = 0.016 \ &f(-2.2) = \log_{10}(-5.6(-2.2) + 1.3) - 2.2 + 1 = \log_{10}(13.62) - 1.2 \approx 1.134 - 1.2 = -0.066 \end{aligned} $$ ## Refine the smaller root Using the *Transcendental Equation Solving* knowledge point $$ \begin{aligned} x &\approx -2.1 \text{ (rounded to the nearest tenth)} \end{aligned} $$ Graph a system of equations to solve $\log (-5.6x + 1.3) = -1 - x$. Round to the nearest tenth. From the least to the greatest, the solutions are: $x \approx$ -2.1 and $x \approx$ 0.2 { "elements": [ { "type": "functiongraph", "params": [ { "js": "Math.log10(-5.6*x + 1.3)", "latex": "\log_{10}(-5.6x + 1.3)" }, -10, 0.23 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = log(-5.6x + 1.3)", "withLabel": true } }, { "type": "functiongraph", "params": [ { "js": "-1 - x", "latex": "-1 - x" }, -10, 10 ], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "name": "y = -1 - x", "withLabel": true } }, { "type": "point", "params": [ [-2.12, 1.12] ], "properties": { "name": "(-2.1, 1.1)", "size": 4, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [ [0.2, -1.2] ], "properties": { "name": "(0.2, -1.2)", "size": 4, "color": "#5583F2", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Solving Exponential Equations Graphically" ] }

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QUESTION IMAGE

Question

Explanation:

Define the system of equations

Determine the domain of the logarithmic function

Find the intersection points numerically

Refine the smaller root

Answer: