solve the inequality \frac{(x + 2)(x - 2)}{x - 7} \ge 0. enter the solution in interval notation.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Interval Notation" ], "new_concepts": [ "Rational Inequalities", "Sign Chart Method" ], "current_concepts": [ "Interval Notation", "Rational Inequalities", "Sign Chart Method" ] } ## Find the critical points of the rational expression $$ \begin{aligned} &\text{Numerator roots: } x + 2 = 0 \implies x = -2 \ &\text{Numerator roots: } x - 2 = 0 \implies x = 2 \ &\text{Denominator roots: } x - 7 = 0 \implies x = 7 \quad (x eq 7) \end{aligned} $$ ## Determine the sign of the expression on each interval $$ \begin{aligned} &(-\infty, -2): \frac{(-)(-)}{(-)} < 0 \ &[-2, 2]: \frac{(+)(-)}{(-)} \ge 0 \ &(2, 7): \frac{(+)(+)}{(-)} < 0 \ &(7, \infty): \frac{(+)(+)}{(+)} > 0 \end{aligned} $$ ## Write the solution set in interval notation $$ x \in [-2, 2] \cup (7, \infty) $$ Solve the inequality $\frac{(x + 2)(x - 2)}{x - 7} \ge 0$. Enter the solution in interval notation. $[-2, 2] \cup (7, \infty)$ { "elements": [ { "type": "bentinequality", "params": [ { "x": -2, "operator": ">=", "showLabel": false } ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3 } }, { "type": "bentinequality", "params": [ { "x": 2, "operator": "<=", "showLabel": false } ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3 } }, { "type": "bentinequality", "params": [ { "x": 7, "operator": ">", "showLabel": false } ], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3 } }, { "type": "point", "params": [[-2, 0]], "properties": { "name": "-2", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[2, 0]], "properties": { "name": "2", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[7, 0]], "properties": { "name": "7", "size": 4, "color": "#5583F2", "fillColor": "white", "withLabel": true } } ], "timestamps": [0.5, 1.0, 1.5] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Rational Inequalities" ] }

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Find the critical points of the rational expression

Determine the sign of the expression on each interval

Write the solution set in interval notation

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