SAT Math Top 10 Questions

Cheatsheet Content

### 1. Solving Linear Equations Master isolating variables. Includes single-variable and systems of equations. **Example 1.1:** If $3x + 5 = 2x - 7$, what is the value of $x$? * **Solution:** Subtract $2x$ from both sides: $x + 5 = -7$. Subtract $5$ from both sides: $x = -12$. **Example 1.2:** Given $2x + y = 10$ and $x - y = 2$, what is the value of $x$? * **Solution:** Add the two equations: $(2x + y) + (x - y) = 10 + 2 \Rightarrow 3x = 12 \Rightarrow x = 4$. **Example 1.3:** If $4(x - 3) = 20$, what is $x$? * **Solution:** Divide by 4: $x - 3 = 5$. Add 3: $x = 8$. **Example 1.4:** If $5x - 2(x + 1) = 13$, what is $x$? * **Solution:** $5x - 2x - 2 = 13 \Rightarrow 3x - 2 = 13 \Rightarrow 3x = 15 \Rightarrow x = 5$. **Example 1.5:** If $\frac{x}{3} + 4 = 7$, what is $x$? * **Solution:** Subtract 4: $\frac{x}{3} = 3$. Multiply by 3: $x = 9$. **Example 1.6:** If $3x - 2y = 12$ and $x = 4$, what is $y$? * **Solution:** Substitute $x=4$: $3(4) - 2y = 12 \Rightarrow 12 - 2y = 12 \Rightarrow -2y = 0 \Rightarrow y = 0$. **Example 1.7:** If $0.5x + 1.5 = 4.0$, what is $x$? * **Solution:** Subtract 1.5: $0.5x = 2.5$. Divide by 0.5: $x = 5$. ### 2. Functions (Evaluating & Interpreting) Understand function notation, domain, range, and interpreting graphs. **Example 2.1:** If $f(x) = 2x^2 - 3x + 1$, what is $f(-2)$? * **Solution:** Substitute $x = -2$: $f(-2) = 2(-2)^2 - 3(-2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15$. **Example 2.2:** For a function $g(x) = ax - 5$, if $g(3) = 7$, what is the value of $a$? * **Solution:** $a(3) - 5 = 7 \Rightarrow 3a = 12 \Rightarrow a = 4$. **Example 2.3:** The graph of $y = h(x)$ contains the point $(-1, 4)$. What is $h(-1)$? * **Solution:** By definition, $h(-1) = 4$. **Example 2.4:** If $f(x) = x - 3$ and $g(x) = x^2$, what is $f(g(2))$? * **Solution:** First find $g(2) = 2^2 = 4$. Then find $f(4) = 4 - 3 = 1$. So, $f(g(2)) = 1$. **Example 2.5:** A function $P(t) = 500(1.03)^t$ models the population of a town after $t$ years. What does 500 represent? * **Solution:** 500 represents the initial population of the town (at $t=0$). **Example 2.6:** If $f(x) = \frac{x + 1}{x - 2}$, for what value of $x$ is $f(x)$ undefined? * **Solution:** A rational function is undefined when its denominator is zero. So, $x - 2 = 0 \Rightarrow x = 2$. **Example 2.7:** Given $f(x) = -x + 4$, what is the $y$-intercept of the graph of $f(x)$? * **Solution:** The $y$-intercept occurs when $x=0$. $f(0) = -0 + 4 = 4$. So the $y$-intercept is $(0, 4)$. ### 3. Percentages, Ratios, and Proportions Crucial for problem-solving in various contexts. **Example 3.1 (Percentage):** A shirt costs $40. If it's on sale for 20% off, what is the new price? * **Solution:** Discount = $0.20 \times 40 = 8$. New price = $40 - 8 = 32$. **Example 3.2 (Ratio):** The ratio of boys to girls in a class is 3:2. If there are 18 boys, how many girls are there? * **Solution:** Let the number of boys be $3k$ and girls be $2k$. $3k = 18 \Rightarrow k = 6$. Number of girls = $2k = 2(6) = 12$. **Example 3.3 (Proportion):** If 5 pencils cost $2.50, how much do 12 pencils cost? * **Solution:** Set up a proportion: $\frac{5}{2.50} = \frac{12}{x}$. Cross-multiply: $5x = 12 \times 2.50 \Rightarrow 5x = 30 \Rightarrow x = 6$. So, 12 pencils cost $6.00. **Example 3.4 (Percentage Increase):** A stock price increased from $50 to $60. What is the percentage increase? * **Solution:** Increase amount = $60 - 50 = 10$. Percentage increase = $\frac{\text{Increase}}{\text{Original}} \times 100\% = \frac{10}{50} \times 100\% = 0.2 \times 100\% = 20\%$. **Example 3.5 (Ratio with Total):** The ratio of cats to dogs at a shelter is 2:5. If there are 35 animals in total, how many dogs are there? * **Solution:** Total parts = $2 + 5 = 7$. Each part represents $\frac{35}{7} = 5$ animals. Number of dogs = $5 \times 5 = 25$. **Example 3.6 (Unit Rate):** A car travels 240 miles on 8 gallons of gas. How many miles per gallon does it get? * **Solution:** Miles per gallon = $\frac{240 \text{ miles}}{8 \text{ gallons}} = 30$ miles/gallon. **Example 3.7 (Reverse Percentage):** A jacket is on sale for $72, which is 25% off its original price. What was the original price? * **Solution:** If it's 25% off, the sale price is 75% of the original price. Let original price be $P$. $0.75P = 72 \Rightarrow P = \frac{72}{0.75} = 96$. The original price was $96. ### 4. Quadratic Equations Solving for roots, understanding parabolas, and the quadratic formula. **Example 4.1:** Solve for $x$: $x^2 - 5x + 6 = 0$. * **Solution:** Factor: $(x - 2)(x - 3) = 0$. So, $x = 2$ or $x = 3$. **Example 4.2:** If $x^2 - 16 = 0$, what are the possible values of $x$? * **Solution:** $x^2 = 16 \Rightarrow x = \pm\sqrt{16} \Rightarrow x = 4$ or $x = -4$. **Example 4.3:** Find the solutions to $x^2 - 4x - 12 = 0$. * **Solution:** Factor: $(x - 6)(x + 2) = 0$. So, $x = 6$ or $x = -2$. **Example 4.4:** For the equation $x^2 + 3x - 1 = 0$, find the solutions using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. * **Solution:** $a=1, b=3, c=-1$. $x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-1)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 4}}{2} = \frac{-3 \pm \sqrt{13}}{2}$. **Example 4.5:** If the vertex of a parabola is $(3, -2)$ and it opens upwards, which of the following could be its equation? A) $y = (x - 3)^2 + 2$ B) $y = (x + 3)^2 - 2$ C) $y = (x - 3)^2 - 2$ D) $y = (x + 3)^2 + 2$ * **Solution:** The vertex form is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. Since it opens upwards, $a > 0$. So, $y = (x - 3)^2 - 2$ is the correct form. Answer C. **Example 4.6:** What is the sum of the roots of $2x^2 + 8x - 5 = 0$? * **Solution:** For a quadratic $ax^2 + bx + c = 0$, the sum of the roots is $-\frac{b}{a}$. Sum = $-\frac{8}{2} = -4$. **Example 4.7:** What is the product of the roots of $3x^2 - 6x + 9 = 0$? * **Solution:** For a quadratic $ax^2 + bx + c = 0$, the product of the roots is $\frac{c}{a}$. Product = $\frac{9}{3} = 3$. ### 5. Data Analysis (Mean, Median, Mode, Standard Deviation) Interpreting statistical measures and data representations (graphs, tables). **Example 5.1:** For the data set $\{2, 4, 4, 6, 9\}$, what is the median? * **Solution:** Arrange in order: $\{2, 4, 4, 6, 9\}$. The middle value is 4. **Example 5.2:** What is the mean of the numbers $\{1, 3, 5, 7, 9\}$? * **Solution:** Mean = $\frac{1+3+5+7+9}{5} = \frac{25}{5} = 5$. **Example 5.3:** In a set of test scores $\{70, 80, 85, 85, 90, 95\}$, what is the mode? * **Solution:** The mode is the number that appears most frequently, which is 85. **Example 5.4:** A data set has a mean of 10 and a standard deviation of 2. If every value in the data set is increased by 5, what are the new mean and standard deviation? * **Solution:** The new mean will be $10 + 5 = 15$. The standard deviation will remain 2 (adding a constant shifts the data but doesn't change its spread). **Example 5.5:** The average (arithmetic mean) of $x$ and $y$ is 10. If $x=6$, what is $y$? * **Solution:** $\frac{x + y}{2} = 10 \Rightarrow x + y = 20$. Substitute $x=6$: $6 + y = 20 \Rightarrow y = 14$. **Example 5.6:** A histogram shows the distribution of ages of people attending a concert. If the bars are skewed to the right, what does this suggest about the ages? * **Solution:** Skewed to the right means there are more younger people, with a few older people causing a "tail" to the right. The median would likely be less than the mean. **Example 5.7:** A student scored 80, 85, and 90 on three tests. What score does the student need on the fourth test to have an average score of 88? * **Solution:** Let the fourth score be $x$. $\frac{80 + 85 + 90 + x}{4} = 88 \Rightarrow \frac{255 + x}{4} = 88 \Rightarrow 255 + x = 352 \Rightarrow x = 97$. ### 6. Geometry: Area, Volume, and Perimeter Formulas for basic shapes (triangles, rectangles, circles, cylinders, cones). **Example 6.1:** What is the area of a circle with a radius of 5? * **Solution:** Area = $\pi r^2 = \pi (5^2) = 25\pi$. **Example 6.2:** A square has a perimeter of 36 units. What is its area? * **Solution:** If perimeter is 36, then each side length $s = \frac{36}{4} = 9$ units. Area = $s^2 = 9^2 = 81$ square units. **Example 6.3:** A rectangular prism has length 6, width 4, and height 3. What is its volume? * **Solution:** Volume = length $\times$ width $\times$ height = $6 \times 4 \times 3 = 72$ cubic units. **Example 6.4:** The circumference of a circle is $10\pi$. What is its radius? * **Solution:** Circumference $C = 2\pi r$. So, $10\pi = 2\pi r \Rightarrow r = 5$. **Example 6.5:** In a right triangle, the two legs are 3 and 4. What is the length of the hypotenuse? * **Solution:** By Pythagorean theorem: $a^2 + b^2 = c^2 \Rightarrow 3^2 + 4^2 = c^2 \Rightarrow 9 + 16 = c^2 \Rightarrow 25 = c^2 \Rightarrow c = 5$. **Example 6.6:** What is the area of a trapezoid with parallel bases of length 6 and 10, and a height of 5? * **Solution:** Area = $\frac{1}{2}(b_1 + b_2)h = \frac{1}{2}(6 + 10)5 = \frac{1}{2}(16)5 = 8 \times 5 = 40$ square units. **Example 6.7:** A cube has a side length of 4 cm. What is its surface area? * **Solution:** A cube has 6 identical square faces. Area of one face = $s^2 = 4^2 = 16$ cm$^2$. Total surface area = $6 \times 16 = 96$ cm$^2$. ### 7. Slope-Intercept Form of a Line Understanding $y = mx + b$, parallel and perpendicular lines. **Example 7.1:** What is the slope of the line passing through $(2, 3)$ and $(4, 7)$? * **Solution:** Slope $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2$. **Example 7.2:** Write the equation of a line with a slope of -3 and a $y$-intercept of 5. * **Solution:** Using $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept: $y = -3x + 5$. **Example 7.3:** What is the slope of a line parallel to $y = \frac{1}{2}x - 4$? * **Solution:** Parallel lines have the same slope. So, the slope is $\frac{1}{2}$. **Example 7.4:** Find the equation of the line that passes through the point $(1, 2)$ and has a slope of 3. * **Solution:** Use point-slope form $y - y_1 = m(x - x_1)$: $y - 2 = 3(x - 1) \Rightarrow y - 2 = 3x - 3 \Rightarrow y = 3x - 1$. **Example 7.5:** What is the $y$-intercept of the line $2x + 3y = 12$? * **Solution:** To find the $y$-intercept, set $x=0$: $2(0) + 3y = 12 \Rightarrow 3y = 12 \Rightarrow y = 4$. The $y$-intercept is $(0, 4)$. **Example 7.6:** Which of the following lines is perpendicular to $y = -2x + 1$? A) $y = 2x + 1$ B) $y = -\frac{1}{2}x + 1$ C) $y = \frac{1}{2}x - 3$ D) $y = -2x - 3$ * **Solution:** The slope of the given line is $-2$. The slope of a perpendicular line is the negative reciprocal, which is $\frac{1}{2}$. So, C is the answer. **Example 7.7:** A line has an $x$-intercept of 3 and a $y$-intercept of 6. What is its slope? * **Solution:** The line passes through $(3, 0)$ and $(0, 6)$. Slope $m = \frac{6 - 0}{0 - 3} = \frac{6}{-3} = -2$. ### 8. Exponents and Radicals Rules for manipulating powers and roots. **Example 8.1:** Simplify $(x^3)^2 \cdot x^4$. * **Solution:** $(x^3)^2 \cdot x^4 = x^{3 \cdot 2} \cdot x^4 = x^6 \cdot x^4 = x^{6+4} = x^{10}$. **Example 8.2:** What is the value of $25^{1/2}$? * **Solution:** $25^{1/2} = \sqrt{25} = 5$. **Example 8.3:** Simplify $\sqrt{72}$. * **Solution:** $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$. **Example 8.4:** If $3^x = 81$, what is the value of $x$? * **Solution:** Since $81 = 3^4$, then $3^x = 3^4 \Rightarrow x = 4$. **Example 8.5:** Express $a^{2/3}$ in radical form. * **Solution:** $a^{2/3} = \sqrt[3]{a^2}$ or $(\sqrt[3]{a})^2$. **Example 8.6:** Simplify $\frac{(2x^2)^3}{4x^4}$. * **Solution:** $\frac{(2x^2)^3}{4x^4} = \frac{2^3 (x^2)^3}{4x^4} = \frac{8x^6}{4x^4} = 2x^{6-4} = 2x^2$. **Example 8.7:** If $x > 0$, which of the following is equivalent to $\frac{1}{\sqrt{x}}$? A) $x^{-1/2}$ B) $x^{1/2}$ C) $x^{-1}$ D) $x^2$ * **Solution:** $\frac{1}{\sqrt{x}} = \frac{1}{x^{1/2}} = x^{-1/2}$. Answer A. ### 9. Word Problems (Translating to Equations) Converting real-world scenarios into mathematical expressions. **Example 9.1:** A farmer has chickens and pigs. There are 20 heads and 56 legs in total. How many chickens does the farmer have? * **Solution:** Let $c$ be chickens and $p$ be pigs. $c + p = 20$ (heads) $2c + 4p = 56$ (legs) From first equation, $p = 20 - c$. Substitute into second: $2c + 4(20 - c) = 56$ $2c + 80 - 4c = 56$ $-2c = 56 - 80$ $-2c = -24 \Rightarrow c = 12$. There are 12 chickens. **Example 9.2:** Emily is 5 years older than twice David's age. If Emily is 25, how old is David? * **Solution:** Let David's age be $D$. Emily's age is $2D + 5$. So, $2D + 5 = 25 \Rightarrow 2D = 20 \Rightarrow D = 10$. David is 10 years old. **Example 9.3:** A store sells apples for $1.00 each and bananas for $0.75 each. If a customer buys 5 apples and some bananas for a total of $9.50, how many bananas did they buy? * **Solution:** Cost of apples = $5 \times 1.00 = 5.00$. Remaining money for bananas = $9.50 - 5.00 = 4.50$. Number of bananas = $\frac{4.50}{0.75} = 6$. **Example 9.4:** The sum of three consecutive integers is 63. What is the smallest integer? * **Solution:** Let the integers be $n, n+1, n+2$. $n + (n+1) + (n+2) = 63$ $3n + 3 = 63$ $3n = 60 \Rightarrow n = 20$. The smallest integer is 20. **Example 9.5:** A car travels at an average speed of 50 mph for 3 hours. How far did it travel? * **Solution:** Distance = Speed $\times$ Time = $50 \text{ mph} \times 3 \text{ hours} = 150$ miles. **Example 9.6:** A rectangular garden has a length that is twice its width. If the perimeter is 36 feet, what are the dimensions of the garden? * **Solution:** Let width be $w$, length be $2w$. Perimeter $2(L+W) = 2(2w + w) = 2(3w) = 6w$. $6w = 36 \Rightarrow w = 6$ feet. Length = $2w = 12$ feet. **Example 9.7:** John has $x$ dollars. Mary has $y$ dollars. If John gives Mary $5, John will have twice as much money as Mary. Which equation represents this? * **Solution:** After the exchange, John has $x-5$ and Mary has $y+5$. $x - 5 = 2(y + 5)$. ### 10. Inequalities Solving and graphing linear and quadratic inequalities. **Example 10.1:** Solve for $x$: $2x - 3 0$? * **Solution:** Factor: $(x - 2)(x + 2) > 0$. Critical points are $x=2$ and $x=-2$. Test intervals: If $x 0$. (Solution) If $-2 2$, e.g., $x=3$: $(1)(5) = 5 > 0$. (Solution) So, the solution is $x 2$. **Example 10.4:** If $3 \le 2x - 1 \le 9$, what is the range of possible values for $x$? * **Solution:** Add 1 to all parts: $4 \le 2x \le 10$. Divide by 2: $2 \le x \le 5$. **Example 10.5:** Which integer is not a solution to $x + 2 > 0$ and $x - 3 0 \Rightarrow x > -2$. $x - 3 10n - 500$. If the company wants to make a profit, what is the minimum number of items they must sell? * **Solution:** $10n - 500 > 0 \Rightarrow 10n > 500 \Rightarrow n > 50$. Since $n$ must be an integer, the minimum is 51 items. **Example 10.7:** If $y > 2x + 1$, which of the following points could be in the solution set? A) $(0, 0)$ B) $(1, 2)$ C) $(1, 4)$ D) $(-1, -1)$ * **Solution:** Test $(1, 4)$: $4 > 2(1) + 1 \Rightarrow 4 > 3$. This is true. So $(1, 4)$ is a solution. ### 11. Complex Numbers Basic operations ($i^2 = -1$). **Example 11.1:** If $i = \sqrt{-1}$, what is the value of $i^7$? * **Solution:** $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. The cycle repeats every 4 powers. $i^7 = i^{4+3} = i^3 = -i$. **Example 11.2:** Simplify $(5 - 2i) - (3 + 4i)$. * **Solution:** $(5 - 3) + (-2 - 4)i = 2 - 6i$. **Example 11.3:** What is the product of $(2 + i)$ and $(3 - i)$? * **Solution:** Use FOIL: $(2)(3) + (2)(-i) + (i)(3) + (i)(-i) = 6 - 2i + 3i - i^2 = 6 + i - (-1) = 6 + i + 1 = 7 + i$. **Example 11.4:** If $a + bi = (3 + 2i)(1 - i)$, what are the values of $a$ and $b$? * **Solution:** $(3 + 2i)(1 - i) = 3(1) + 3(-i) + 2i(1) + 2i(-i) = 3 - 3i + 2i - 2i^2 = 3 - i - 2(-1) = 3 - i + 2 = 5 - i$. So $a=5, b=-1$. **Example 11.5:** What is the conjugate of $7 - 3i$? * **Solution:** The conjugate of $a + bi$ is $a - bi$. So the conjugate is $7 + 3i$. **Example 11.6:** If $(x + yi)(1 + i) = 4 + 2i$, where $x$ and $y$ are real numbers, what is the value of $x$? * **Solution:** Expand the left side: $x(1) + x(i) + yi(1) + yi(i) = x + xi + yi + yi^2 = x + xi + yi - y = (x - y) + (x + y)i$. So, $(x - y) + (x + y)i = 4 + 2i$. Equate real and imaginary parts: $x - y = 4$ $x + y = 2$ Add the two equations: $(x - y) + (x + y) = 4 + 2 \Rightarrow 2x = 6 \Rightarrow x = 3$. **Example 11.7:** If $i^n = -1$, which of the following could be $n$? A) 1 B) 2 C) 3 D) 4 * **Solution:** $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. So, $n=2$. Answer B. ### 12. Polynomials (Operations & Factoring) Adding, subtracting, multiplying, dividing, and factoring polynomials. **Example 12.1:** Expand $(x + 5)^2$. * **Solution:** $(x + 5)(x + 5) = x^2 + 5x + 5x + 25 = x^2 + 10x + 25$. **Example 12.2:** Factor $x^2 + 2x - 15$. * **Solution:** Find two numbers that multiply to -15 and add to 2. These are 5 and -3. So, $(x + 5)(x - 3)$. **Example 12.3:** What is the result of $(3x^2 + 2x - 1) - (x^2 - 4x + 3)$? * **Solution:** $(3x^2 - x^2) + (2x - (-4x)) + (-1 - 3) = 2x^2 + 6x - 4$. **Example 12.4:** If $(x - 1)$ is a factor of $x^3 - 2x^2 + kx + 4$, what is the value of $k$? * **Solution:** If $(x - 1)$ is a factor, then $x=1$ is a root, so $f(1) = 0$. $1^3 - 2(1)^2 + k(1) + 4 = 0$ $1 - 2 + k + 4 = 0$ $3 + k = 0 \Rightarrow k = -3$. **Example 12.5:** Factor $4x^2 - 9$. * **Solution:** This is a difference of squares: $(2x)^2 - 3^2 = (2x - 3)(2x + 3)$. **Example 12.6:** What is the product of $(x + 2)$ and $(x^2 - 2x + 4)$? * **Solution:** This is the sum of cubes formula $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. Here $a=x, b=2$. So, $(x + 2)(x^2 - 2x + 4) = x^3 + 2^3 = x^3 + 8$. (Alternatively, multiply it out: $x(x^2 - 2x + 4) + 2(x^2 - 2x + 4) = x^3 - 2x^2 + 4x + 2x^2 - 4x + 8 = x^3 + 8$). **Example 12.7:** If $P(x) = x^3 - 5x^2 + 2x - 10$, what is the value of $P(5)$? * **Solution:** $P(5) = 5^3 - 5(5^2) + 2(5) - 10 = 125 - 5(25) + 10 - 10 = 125 - 125 + 0 = 0$. ### 13. Radical and Rational Equations Solving equations involving square roots or fractions with variables in the denominator. **Example 13.1:** Solve for $x$: $\sqrt{x + 5} = 3$. * **Solution:** Square both sides: $(\sqrt{x + 5})^2 = 3^2 \Rightarrow x + 5 = 9 \Rightarrow x = 4$. **Example 13.2:** Solve for $x$: $\frac{3}{x} = \frac{1}{4}$. * **Solution:** Cross-multiply: $3 \times 4 = 1 \times x \Rightarrow 12 = x$. **Example 13.3:** Solve for $x$: $\frac{x}{x - 2} = 5$. * **Solution:** Multiply both sides by $(x - 2)$: $x = 5(x - 2) \Rightarrow x = 5x - 10 \Rightarrow -4x = -10 \Rightarrow x = \frac{10}{4} = \frac{5}{2}$. (Check: denominator $x-2 = \frac{5}{2}-2 = \frac{1}{2} \ne 0$). **Example 13.4:** Solve for $x$: $\sqrt{2x - 1} + 2 = 5$. * **Solution:** Subtract 2: $\sqrt{2x - 1} = 3$. Square both sides: $2x - 1 = 9 \Rightarrow 2x = 10 \Rightarrow x = 5$. **Example 13.5:** Solve for $x$: $\frac{1}{x} + \frac{1}{2x} = \frac{3}{4}$. * **Solution:** Find a common denominator, $4x$. Multiply the entire equation by $4x$: $4x(\frac{1}{x}) + 4x(\frac{1}{2x}) = 4x(\frac{3}{4})$ $4 + 2 = 3x$ $6 = 3x \Rightarrow x = 2$. **Example 13.6:** If $\frac{x + 1}{x - 1} = \frac{3}{2}$, what is $x$? * **Solution:** Cross-multiply: $2(x + 1) = 3(x - 1) \Rightarrow 2x + 2 = 3x - 3 \Rightarrow 5 = x$. **Example 13.7:** What is the solution to $\sqrt{x} = x - 2$? * **Solution:** Square both sides: $x = (x - 2)^2 \Rightarrow x = x^2 - 4x + 4$. Rearrange into a quadratic: $x^2 - 5x + 4 = 0$. Factor: $(x - 1)(x - 4) = 0$. So, $x = 1$ or $x = 4$. **Check for extraneous solutions:** For $x=1$: $\sqrt{1} = 1$, and $1 - 2 = -1$. Since $1 \ne -1$, $x=1$ is an extraneous solution. For $x=4$: $\sqrt{4} = 2$, and $4 - 2 = 2$. Since $2 = 2$, $x=4$ is a valid solution. The only solution is $x = 4$. ### 14. Systems of Inequalities Graphing and finding solution regions for multiple inequalities. **Example 14.1:** Which point is a solution to the system $y > x + 1$ and $y 0 + 1$ (True, $2 > 1$) and $2 -2x + 5$, which of the following points is a solution? A) $(2, 0)$ B) $(0, 0)$ C) $(3, 1)$ D) $(1, 1)$ * **Solution:** Test $(3, 1)$: $1 -2(3) + 5 \Rightarrow 1 > -6 + 5 \Rightarrow 1 > -1$ (True) So, $(3, 1)$ is a solution. **Example 14.7:** A student is buying notebooks and pens. Notebooks cost $3 each and pens cost $1 each. The student has a budget of $10 and wants to buy at least 2 notebooks. If $n$ is the number of notebooks and $p$ is the number of pens, write a system of inequalities representing this situation. * **Solution:** $3n + p \le 10$ (budget constraint) $n \ge 2$ (at least 2 notebooks) $p \ge 0$ (cannot buy negative pens) $n$ must be an integer, $p$ must be an integer. ### 15. Trigonometry Basics SOH CAH TOA, sine, cosine, tangent in right triangles. **Example 15.1:** In a right triangle, if the opposite side to angle $\theta$ is 3 and the hypotenuse is 5, what is $\sin(\theta)$? * **Solution:** $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5}$. **Example 15.2:** If $\cos(\theta) = \frac{4}{5}$ in a right triangle, what is $\tan(\theta)$? * **Solution:** If adjacent = 4, hypotenuse = 5, then by Pythagorean theorem, opposite = $\sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3$. So, $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{3}{4}$. **Example 15.3:** In a right triangle, if an angle is $30^\circ$ and the hypotenuse is 10, what is the length of the side opposite the $30^\circ$ angle? * **Solution:** Let the opposite side be $x$. $\sin(30^\circ) = \frac{x}{10}$. Since $\sin(30^\circ) = \frac{1}{2}$, we have $\frac{1}{2} = \frac{x}{10} \Rightarrow 2x = 10 \Rightarrow x = 5$. **Example 15.4:** In a right triangle, if $\sin(A) = \frac{5}{13}$, what is $\cos(B)$ where $A$ and $B$ are the acute angles of the triangle? * **Solution:** In a right triangle, the sine of one acute angle is equal to the cosine of the other acute angle. So, $\cos(B) = \sin(A) = \frac{5}{13}$. **Example 15.5:** An observer stands 100 feet from the base of a tree. The angle of elevation to the top of the tree is $45^\circ$. What is the height of the tree? * **Solution:** Let $h$ be the height of the tree. $\tan(45^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{100}$. Since $\tan(45^\circ) = 1$, we have $1 = \frac{h}{100} \Rightarrow h = 100$ feet. **Example 15.6:** If $\sin(x) = 0.6$ and $x$ is an acute angle, what is $\cos(x)$? * **Solution:** Use the identity $\sin^2(x) + \cos^2(x) = 1$. $(0.6)^2 + \cos^2(x) = 1$ $0.36 + \cos^2(x) = 1$ $\cos^2(x) = 1 - 0.36 = 0.64$ $\cos(x) = \sqrt{0.64} = 0.8$ (since $x$ is acute, $\cos(x)$ is positive). **Example 15.7:** In a right triangle, the adjacent side to angle $\theta$ is 8 and the hypotenuse is 17. What is $\cos(\theta)$? * **Solution:** $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{8}{17}$.