15) solve the following inequalities: a) $ \frac{2}{3}(2 - x) \frac{1}{2} + x $ b) $ x^2 - 16 3x - 2 $ 16) for the following oblique triangles, note that the placement test is no - calculator, so you may be asked to identify which formula (sine or cosine law) you would use, how you would manipulate the formula, and how to substitute numbers up to, but not including, any step that would require a calculator. (the answer key to this guide will show the last step before a calculator would be used, but will also show the final answer from a calculator so you can check your final work.) a) if $ a = 6.2 $ cm, $ a = 65.0^\circ $, and $ b = 55.2^\circ $, solve for side $ c $ to the nearest tenth. b) if $ a = 21.61 $ cm, $ b = 29.33 $ cm, and $ c = 42.57 $ cm, solve for the smallest angle in the triangle to the nearest tenth. math placement test study guide | grade 11 (into math 182) exercises grade 11 mathematics content – for placement into grade 12 math (math 30 - 1/math 182) answer key

Question

15) solve the following inequalities:
    a) $ \frac{2}{3}(2 - x) > \frac{1}{2} + x $
    b) $ x^2 - 16 < 6x $
    c) draw the graph of the solution to the following, including shading and correct solid vs. dashed line: $ y > 3x - 2 $
16) for the following oblique triangles, note that the placement test is no - calculator, so you may be asked to identify which formula (sine or cosine law) you would use, how you would manipulate the formula, and how to substitute numbers up to, but not including, any step that would require a calculator. (the answer key to this guide will show the last step before a calculator would be used, but will also show the final answer from a calculator so you can check your final work.)
    a) if $ a = 6.2 $ cm, $ a = 65.0^\circ $, and $ b = 55.2^\circ $, solve for side $ c $ to the nearest tenth.
    b) if $ a = 21.61 $ cm, $ b = 29.33 $ cm, and $ c = 42.57 $ cm, solve for the smallest angle in the triangle to the nearest tenth.
math placement test study guide | grade 11 (into math 182) exercises
grade 11 mathematics content – for placement into
grade 12 math (math 30 - 1/math 182)
answer key

Accepted Answer

### 15a) Solve $\boldsymbol{\frac{2}{3}(2 - x)>\frac{1}{2}+x}$ # Explanation: ## Step1: Eliminate denominators (LCM of 3,2 is 6) Multiply both sides by 6: $6\times\frac{2}{3}(2 - x)>6\times\frac{1}{2}+6x$ Simplify: $4(2 - x)>3 + 6x$ ## Step2: Distribute left side $8 - 4x>3 + 6x$ ## Step3: Move variables to one side Add $4x$ to both sides: $8>3 + 10x$ ## Step4: Isolate $x$ Subtract 3: $5>10x$ Divide by 10: $\frac{5}{10}>x$ Simplify: $x<\frac{1}{2}$ # Answer: $x < \frac{1}{2}$ ### 15b) Solve $\boldsymbol{x^2 - 16<6x}$ # Explanation: ## Step1: Rearrange to standard quadratic $x^2 - 6x - 16<0$ ## Step2: Factor the quadratic Find roots of $x^2 - 6x - 16 = 0$. Factor: $(x - 8)(x + 2)=0$ Roots: $x = 8$ or $x=-2$ ## Step3: Analyze sign of $y=x^2 - 6x - 16$ The parabola opens upward (coefficient of $x^2$ is positive). Inequality $y<0$ holds between the roots. # Answer: $-2 < x < 8$ ### 15c) Graph $\boldsymbol{y>3x - 2}$ # Explanation: ## Step1: Identify line type Inequality is $y>3x - 2$ (dashed line, since $>$ not $\geq$). ## Step2: Graph the line $y = 3x - 2$ - Slope $m = 3$, y-intercept $b=-2$ (plot $(0, -2)$, then rise 3, run 1 to plot $(1, 1)$, etc.). - Draw a **dashed** line through these points. ## Step3: Shade the region Test a point (e.g., $(0,0)$): $0>3(0)-2$ (True). Shade the region **above** the dashed line (where $y>3x - 2$ holds). # Answer: Dashed line $y = 3x - 2$, shade above the line. ### 16a) Solve for side $c$ (oblique triangle, $a = 6.2$ cm, $A = 65.0^\circ$, $B = 55.2^\circ$) # Explanation: ## Step1: Find angle $C$ Sum of angles: $C=180^\circ - A - B=180 - 65 - 55.2 = 59.8^\circ$ ## Step2: Apply Law of Sines $\frac{a}{\sin A}=\frac{c}{\sin C}$ Substitute: $\frac{6.2}{\sin 65.0^\circ}=\frac{c}{\sin 59.8^\circ}$ ## Step3: Solve for $c$ $c=\frac{6.2\times\sin 59.8^\circ}{\sin 65.0^\circ}$ (Last step before calculator: $c=\frac{6.2\times\sin 59.8^\circ}{\sin 65.0^\circ}$; calculator gives $c\approx5.9$ cm) # Answer: $c\approx5.9$ cm (process: Law of Sines, $c=\frac{6.2\sin59.8^\circ}{\sin65.0^\circ}$) ### 16b) Solve for the smallest angle ($a = 21.61$ cm, $b = 29.33$ cm, $c = 42.57$ cm) # Explanation: ## Step1: Identify smallest angle Smallest angle is opposite the shortest side ($a = 21.61$ cm, opposite angle $A$). ## Step2: Apply Law of Cosines $\cos A=\frac{b^2 + c^2 - a^2}{2bc}$ ## Step3: Substitute values $\cos A=\frac{(29.33)^2+(42.57)^2-(21.61)^2}{2\times29.33\times42.57}$ (Last step before calculator: compute numerator/denominator; calculator gives $A\approx25.2^\circ$) # Answer: Smallest angle $\approx25.2^\circ$ (process: Law of Cosines, $\cos A=\frac{29.33^2 + 42.57^2 - 21.61^2}{2\times29.33\times42.57}$)

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QUESTION IMAGE

Question

Explanation:

15a) Solve \(\boldsymbol{\frac{2}{3}(2 - x)>\frac{1}{2}+x}\)

Step1: Eliminate denominators (LCM of 3,2 is 6)

Step2: Distribute left side

Step3: Move variables to one side

Step4: Isolate \(x\)

Step1: Rearrange to standard quadratic

Step2: Factor the quadratic

Step3: Analyze sign of \(y=x^2 - 6x - 16\)

Step1: Identify line type

Step2: Graph the line \(y = 3x - 2\)

Step3: Shade the region

Answer:

15b) Solve \(\boldsymbol{x^2 - 16<6x}\)