lesson 10: equations, formulas, and inequalities

solve the following equations and inequalities for the unknown variable. round answers to two (2) decimal places if necessary.

10. $5n + 2 = 27$

11. $\frac{1}{3}(6 + x) - 5 = 12$

12. $\frac{(3x)2 - 10}{4} = 56$

13. $\sqrt{4x} + 14 = 32$

14. $3x^2 + 2(x^2 - 20) = 16 - 2x^2$

15. $\frac{1}{3}x + 2 \ge 6$

Question

lesson 10: equations, formulas, and inequalities

solve the following equations and inequalities for the unknown variable. round answers to two (2) decimal places if necessary.

10. $5n + 2 = 27$

11. $\frac{1}{3}(6 + x) - 5 = 12$

12. $\frac{(3x)2 - 10}{4} = 56$

13. $\sqrt{4x} + 14 = 32$

14. $3x^2 + 2(x^2 - 20) = 16 - 2x^2$

15. $\frac{1}{3}x + 2 \ge 6$

Accepted Answer

{ "quality": "clear", "question_count": 6, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Basic Arithmetic Operations" ], "new_concepts": [ "Linear Equations Solving", "Radical Equations Solving", "Quadratic Equations Solving" ], "current_concepts": [ "Basic Arithmetic Operations", "Linear Equations Solving", "Radical Equations Solving", "Quadratic Equations Solving", "Linear Inequalities" ] } ## Solve linear equations (10, 11, 12) $$ \begin{aligned} 10.\quad &5n + 2 = 27 \implies 5n = 25 \implies n = 5 \ 11.\quad &\frac{1}{3}(6 + x) - 5 = 12 \implies \frac{1}{3}(6 + x) = 17 \implies 6 + x = 51 \implies x = 45 \ 12.\quad &\frac{(3x)2 - 10}{4} = 56 \implies 6x - 10 = 224 \implies 6x = 234 \implies x = 39 \end{aligned} $$ ## Solve radical and quadratic equations (13, 14) $$ \begin{aligned} 13.\quad &\sqrt{4x} + 14 = 32 \implies \sqrt{4x} = 18 \implies 4x = 324 \implies x = 81 \ 14.\quad &3x^2 + 2(x^2 - 20) = 16 - 2x^2 \implies 3x^2 + 2x^2 - 40 = 16 - 2x^2 \ &5x^2 - 40 = 16 - 2x^2 \implies 7x^2 = 56 \implies x^2 = 8 \implies x = \pm\sqrt{8} \approx \pm 2.83 \end{aligned} $$ ## Solve linear inequality (15) $$ \begin{aligned} 15.\quad &\frac{1}{3}x + 2 \ge 6 \implies \frac{1}{3}x \ge 4 \implies x \ge 12 \end{aligned} $$ | No. | Answer | |---|---| | 10 | $n = 5$ | | 11 | $x = 45$ | | 12 | $x = 39$ | | 13 | $x = 81$ | | 14 | $x = \pm\sqrt{8} \approx \pm 2.83$ | | 15 | $x \ge 12$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Linear Equations Solving" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Solve linear equations (10, 11, 12)

Solve radical and quadratic equations (13, 14)

Solve linear inequality (15)

Answer:

No.	Answer
11	\(x = 45\)
12	\(x = 39\)
13	\(x = 81\)
14	\(x = \pm\sqrt{8} \approx \pm 2.83\)
15	\(x \ge 12\)