Digital SAT Math Practice Question #1187
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Before you begin solving, what values of \( x \) would make any of the denominators equal to zero? These values must be excluded from your potential solutions.
After combining the fractions and simplifying the equation, carefully check if your derived solution(s) are among the values you identified as restricted. What does it mean if they are?
⚡ **Graphing Calculator Approach (Desmos/TI-84):** 1. Rewrite the equation by moving all terms to one side: \( \frac{2x}{x^2-1} + \frac{1}{x+1} - \frac{1}{x-1} = 0 \). 2. Define a function \( y = \frac{2x}{x^2-1} + \frac{1}{x+1} - \frac{1}{x-1} \). 3. Graph this function on a graphing calculator (e.g., Desmos or a TI-84). 4. Observe the graph. You will notice that the graph has vertical asymptotes at \( x=1 \) and \( x=-1 \), but it never crosses or touches the x-axis. This indicates that there are no real values of \( x \) for which \( y=0 \). 5. Since there are no solutions to the equation, the set of values of \( x \) that satisfy the equation is empty. The sum of an empty set is conventionally \( 0 \).
1. **Identify Restrictions:** The first crucial step is to determine the values of \( x \) for which the denominators of the fractions would be zero, as these values are undefined for the original equation. The denominators are \( x^2-1 \),...
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