\(8 - 2x + 8 \ge 3x + 3 + 3 \implies 16 - 2x \ge 3x + 6\)

Subtract \(3x\) and \(16\): \(-5x \ge -10\). Divide by \(-5\) and **flip the inequality sign**: \(x \le 2\).

Question 8: System of Linear Equations CORRECT ANSWER: A

Multiply Eq(1) by 3 and Eq(2) by 2 to eliminate the \(y\)-variable via addition:

\((9x - 6y = 36) + (10x + 6y = 2) \implies 19x = 38 \implies x = 2\)

Plug \(x = 2\) back into Eq(1): \(3(2) - 2y = 12 \implies 6 - 2y = 12 \implies -2y = 6 \implies y = -3\). Solution is \((2, -3)\).

Question 9: Simplifying Rational Algebra CORRECT ANSWER: C

Multiply by the reciprocal and factor each quadratic completely:

\(\frac{(x-7)(x+2)}{(x-2)(x+2)} \cdot \frac{(x-2)(x-3)}{(x-7)(x-3)}\)

All corresponding binomial factors cancel out from numerator and denominator, leaving exactly \(1\).

Question 10: Polynomial Factoring & Geometry CORRECT ANSWER: A

Since \(\text{Area} = \text{Length} \times \text{Width}\), find the missing factor of \(2x^2 + 11x - 21\) given that one factor is \((x + 7)\):

Factor \(2x^2 + 11x - 21\) into binomials: \((2x - 3)(x + 7)\). The missing length is \(2x - 3\).

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Domain 3: Geometric & Spatial Reasoning (Questions 11-15)

Question 11: Distance & Diameter CORRECT ANSWER: A

Find the diameter length using the distance formula between \(A(-3, 5)\) and \(B(5, -1)\):

\(D = \sqrt{(5 - (-3))^2 + (-1 - 5)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = 10\)

The radius of the circle is half of the diameter length: \(R = \frac{D}{2} = \frac{10}{2} = 5\).

Question 12: Composite 3D Geometry CORRECT ANSWER: A

Calculate the cylinder volume and subtract the volume of the carved-out cone:

\(V_{\text{cyl}} = \pi \cdot 5^2 \cdot 12 = 300\pi \quad \text{and} \quad V_{\text{cone}} = \frac{1}{3}\pi \cdot 5^2 \cdot 6 = 50\pi\)

Subtract the cone volume from the cylinder: \(300\pi - 50\pi = 250\pi \text{ cubic inches}\).

Question 13: Right Triangle Trigonometry CORRECT ANSWER: C

Since \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{15}{17}\), use the Pythagorean Theorem to find the opposite side:

\(\text{Opp} = \sqrt{17^2 - 15^2} = \sqrt{289 - 225} = \sqrt{64} = 8\)

Now find the tangent ratio: \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{8}{15}\).

Question 14: Similarity & Scaled Proportions CORRECT ANSWER: B

Set up a proportion of corresponding sides of similar triangles \(\triangle ABC \sim \triangle DEF\):

\(\frac{DE}{AB} = \frac{EF}{BC} \implies \frac{18}{8} = \frac{EF}{12}\)

Cross-multiply and solve: \(8 \cdot EF = 216 \implies EF = 27\text{ cm}\).

Question 15: Perpendicular Coordinate Lines CORRECT ANSWER: A

Express \(2x - 3y = 12\) in slope-intercept form: \(y = \frac{2}{3}x - 4\). Slope is \(m_1 = \frac{2}{3}\).

The slope of a perpendicular line is the negative reciprocal: \(m_2 = -\frac{3}{2}\). Use point-slope form with \((4, -3)\):

\(y - (-3) = -\frac{3}{2}(x - 4) \implies y + 3 = -\frac{3}{2}x + 6 \implies y = -\frac{3}{2}x + 3\)

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Domain 4: Probabilistic & Statistical Reasoning (Questions 16-20)

Question 16: Mean & Median Calculations CORRECT ANSWER: B

First, calculate the arithmetic mean of the five values:

\(\text{Mean} = \frac{78 + 85 + 92 + 85 + 90}{5} = \frac{430}{5} = 86\)

Arrange in order: \(78, 85, 85, 90, 92\). The median (middle value) is \(85\).

Question 17: Interpreting Standard Deviation CORRECT ANSWER: C

Standard deviation represents the dispersion or spread of data points about the mean. A larger standard deviation (\(7.8 > 3.4\)) indicates that Class B's quiz scores are more widely dispersed and exhibit greater variability than Class A's scores.

Question 18: Compound Dependent Probability CORRECT ANSWER: B

Calculate probabilities for each dependent pick in sequence without replacement:

\(P(\text{Draw 1 is Blue}) = \frac{3}{8} \quad \text{and} \quad P(\text{Draw 2 is Blue}) = \frac{2}{7}\)

Multiply the dependent events: \(\frac{3}{8} \times \frac{2}{7} = \frac{6}{56} = \frac{3}{28}\).

Question 19: Fundamental Counting Principle CORRECT ANSWER: A

Determine the number of valid choices for each of the four passcode slots:

• Slot 1: Cannot be 0 or 1 \(\implies \{2, 3, 4, 5, 6, 7, 8, 9\} = 8\text{ choices}\)
• Slot 2 & Slot 3: Any digit is allowed \(\implies 10 \times 10 = 100\text{ choices}\)
• Slot 4: Must be odd \(\implies \{1, 3, 5, 7, 9\} = 5\text{ choices}\)

Total combinations = \(8 \times 10 \times 10 \times 5 = 4,000\text{ passcodes}\).

Question 20: Conditional Probability Table CORRECT ANSWER: B

Calculate \(P(11\text{th Grade} \mid \text{Soccer})\) from the given row totals:

\(\text{Total Soccer Players} = 15 + 18 + 12 + 10 = 55\)

There are \(12\) Soccer players in 11th Grade. Thus, the conditional probability is \(\frac{12}{55}\).

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TSIA2 Core Testing Strategies

1. Capitalize on the Untimed Format

The TSIA2 Math test has no time limits. Encourage students to check their work backwards (plugging choices back into equations) to verify accuracy. Never rush.

2. Master the Negative Sign in Inequalities

One of the most common pitfalls is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. Standard testing questions directly exploit this misconception.

3. Simplify Before You Calculate

In Algebraic and Geometric reasoning, equations and proportions often feature terms that simplify naturally. Students should factor first rather than expand large polynomial products immediately.

4. Read Probabilistic Given Conditions Carefully

Conditional probability ("given that...") changes the denominator of your probability ratio. Identify the sub-population of the sample space before calculating fractions.

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