d $|x - 1| 2|x - 4|$

Question

d $|x - 1| > 2|x - 4|$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Absolute Value Inequalities", "Algebraic Inequalities" ], "new_concepts": [], "current_concepts": [ "Absolute Value Inequalities", "Algebraic Inequalities", "Quadratic Inequalities" ] } ## Square both sides of the inequality $$ \begin{aligned} |x - 1|^2 &> (2|x - 4|)^2 \ (x - 1)^2 &> 4(x - 4)^2 \end{aligned} $$ ## Expand and simplify the quadratic terms $$ \begin{aligned} x^2 - 2x + 1 &> 4(x^2 - 8x + 16) \ x^2 - 2x + 1 &> 4x^2 - 32x + 64 \ 0 &> 3x^2 - 30x + 63 \ 3x^2 - 30x + 63 &< 0 \end{aligned} $$ ## Solve the quadratic inequality $$ \begin{aligned} x^2 - 10x + 21 &< 0 \ (x - 3)(x - 7) &< 0 \ 3 < x &< 7 \end{aligned} $$ Solve the inequality: $|x - 1| > 2|x - 4|$ is $3 < x < 7$ { "elements": [ { "type": "functiongraph", "params": [ { "js": "Math.abs(x - 1)", "latex": "|x - 1|" }, -2, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = |x - 1|", "withLabel": true } }, { "type": "functiongraph", "params": [ { "js": "2 * Math.abs(x - 4)", "latex": "2|x - 4|" }, -2, 10 ], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "name": "y = 2|x - 4|", "withLabel": true } }, { "type": "point", "params": [ [3, 2] ], "properties": { "name": "A(3, 2)", "color": "#5583F2", "size": 3, "withLabel": true } }, { "type": "point", "params": [ [7, 6] ], "properties": { "name": "B(7, 6)", "color": "#5583F2", "size": 3, "withLabel": true } }, { "type": "segment", "params": [ [3, 0], [7, 0] ], "properties": { "strokeColor": "#55DDF2", "strokeWidth": 4, "name": "Solution: 3 < x < 7", "withLabel": true } } ], "timestamps": [0.5, 1.0, 1.5] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Absolute Value Inequalities" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Square both sides of the inequality

Expand and simplify the quadratic terms

Solve the quadratic inequality

Answer: