MIDTERM REVIEW

Basic

1. Basic Linear Equations

Solve for \(x\): \(3x + 7 = 22\)

Solution

\[\eqalign{3x + 7 &=& 22\cr 3x &=& 15\cr x &=& 5\cr }\]

2. Rearranging

Solve for \(y\): \(5 - 2y = 19\)

Solution

\[\eqalign{ 5 - 2y &=& 19\cr -2y &=& 14\cr y &=& -7\cr }\]

3. Distributing values

Solve for \(a\): \(4(a - 3) = 20\)

Solution

\[\eqalign{ 4(a - 3) &=& 20\cr a - 3 &=& 5\cr a &=& 8\cr}\]

4. Gathering Terms

Solve for \(x\): \(7x - 4 = 3x + 12\)

Solution

\[\eqalign{ 7x - 4 &=& 3x + 12\cr 4x - 4 &=& 12\cr 4x &=& 16\cr x &=& 4\cr }\]

5. Fraction

Solve for \(m\): \(\frac{m}{4} + 6 = 11\)

Solution

\[\eqalign{ \frac{m}{4} + 6 &=& 11\cr \frac{m}{4} &=& 5\cr m &=& 20\cr }\]

6. Gathering terms of Intermediate Equations

Solve for \(p\): \(2(3p + 1) - 4 = 5(p - 2)\)

Solution

\[\eqalign{ 2(3p + 1) - 4 &=& 5(p - 2)\cr 6p + 2 - 4 &=& 5p - 10\cr 6p - 2 &=& 5p - 10\cr p &=& -8\cr}\]

7. Fraction

Solve for \(x\): \(\frac{2x + 5}{3} = 7\)

Solution

\[\eqalign{ \frac{2x + 5}{3} = 7\cr 2x + 5 = 21\cr 2x = 16\cr x = 8\cr}\]

8. Inequalities

Solve the inequality for \(x\): \(3x - 8 > 7\)

Solution

\[\eqalign{ 3x - 8 > 7\cr x > 15\cr x > 5\cr}\]

9. Inequalities

Solve the inequality for \(y\): \(-4y + 2 \le 14\)

Solution

\[\eqalign{ -4y + 2 \le 14\cr -4y \le 12\cr y \ge -3\cr }\]

10. Absolute value function

Solve for \(x\): \(\vert{}2x - 3\vert{} = 9\)

Solution \[\vert{}2x - 3\vert{} = 9\]

• Case 1:

\[\eqalign{ 2x - 3 &=& 9\cr 2x &=& 12\cr x &=& 6\cr }\]

• Case 2:

\[\eqalign{ 2x - 3 &=& -9\cr 2x &=& -6\cr x &=& -3\cr }\]

11. Systems of Equations

Solve: \(x + y = 10\) and \(2x - y = 8\)

Solution:

\[\eqalign{ x + y &=& 10\cr 2x - y &=& 8\cr \hline (x + y) + (2x - y) &=& 10 + 8\cr 3x &=& 18\cr x&=& 6\cr \hline (6)+ y &=& 10\cr y &=& 4\cr }\]

12. System of equations

Solve: \(3x + 2y = 12\) and \(5x - 2y = 4\)

Solution:

\[\eqalign{ 3x + 2y &=& 12\cr 5x - 2y &=& 4\cr \hline (3x + 2y) + (5x - 2y) &=& 12 + 4\cr 8x &=& 16\cr x &=& 2\cr \hline 3(2) + 2y &=& 12\cr 6 + 2y &=& 12\cr 2y &=& 6\cr y &=& 3\cr }\]

13. System of equations

Solve: \(4x - y = 7\) and \(2x + 3y = 21\)

Solution:

\[\eqalign{ 4x - y &=& 7\cr 2x + 3y &=& 21\cr \hline 12x - 3y &=& 21\cr (12x - 3y) + (2x + 3y) &=& 21 + 21 \cr 14x &=& 42\cr x &=& 3\cr \hline 4(3) - y &=& 7\cr 12 - y &=& 7\cr y &=& 5\cr \hline (x, y) &=& (3, 5)\cr }\]

14. Exponents & Polynomials

Simplify: \((3x^2y^3)(4x^3y^5)\)

Solution:

\[\matrix{ (3x^2 y^3)(4x^3 y^5)\cr (3 \cdot 4) \cdot x^{2+3} \cdot y^{3+5}\cr 12x^5y^8\cr }\]

15. Combining terms

Simplify: \(\frac{18a^5b^2}{6a^2b^4}\)

Solution:

\[\matrix{ \frac{18a^5b^2}{6a^2b^4}\cr \frac{18}{6} \cdot a^{5-2} \cdot b^{2-4}\cr \frac{3a^3}{b^2}\cr }\]

16. Expansion

Problem Expand and simplify: \((2x - 5)(x + 4)\)

Solution

\[\matrix{ (2x - 5)(x + 4)\cr 2x(x) + 2x(4) - 5(x) - 5(4)\cr 2x^2 + 8x - 5x - 20\cr 2x^2 + 3x - 20\cr }\]

17. Factoring

Problem Factor completely: \(x^2 - 9x + 20\)

Solution

\[\matrix{ x^2 - 9x + 20\cr \hline (x+a)(x+b)\cr x^2+ (a+b)x + a\cdot b\cr \hline a\cdot b = 20\cr a + b = 9\cr a = -4\cr b=-5\cr \hline (x - 4)(x - 5)\cr }\]

18. Difference of Squares

Factor completely: \(4x^2 - 25\)

Solution:

\[\matrix{4x^2 - 25\cr (2x)^2 - (5)^2\cr (2x)^2 + 10x -10x - 5^2\cr (2x - 5)(2x + 5)\cr }\]

19. Quadratic Equations

Solve for \(x\) by factoring: \(x^2 - 6x + 8 = 0\)

Solution:

\[\eqalign{ x^2 - 6x + 8 &=& 0\cr x+ (-2-4)x + (-2)(-4)&=& 0\cr (x - 2)(x - 4) &=& 0\cr x&=&(2,4)\cr }\]

20. Quadratic formula

Solve for \(x\) using the quadratic formula:

\[2x^2 + 5x - 3 = 0\]

Solution:

\[\eqalign{0&=& 2x^2 + 5x - 3\cr \hline &&(a = 2, b = 5, c = -3)\cr x &=& \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \cr &=& \frac{-5 \pm \sqrt{5^2 - 4(2)(-3)}}{2(2)}\cr &=& \frac{-5 \pm \sqrt{25 + 24}}{4}\cr &=& \frac{-5 \pm \sqrt{49}}{4}\cr &=& \frac{-5 \pm 7}{4}\cr \hline x_1 &=& \frac{2}{4}\cr &=& \frac{1}{2}\cr x_2 &=&\frac{-12}{4}\cr &=& -3\cr \hline x &=& \left(\frac{1}{2},-3\right)\cr }\]

Question 6: Logarithms and Exponents

21: Exponential Equations with Different Bases.

Problem: Solve for \(x\):

\[3^{2x+1} = 5^{x-2}\]

Solution:

\[\eqalign{ 3^{2x+1} &=& 5^{x-2}\cr \ln(3^{2x+1}) &=& \ln(5^{x-2})\cr (2x + 1) \ln 3 &=& (x - 2) \ln 5\cr 2x \ln 3 + \ln 3 &=& x \ln 5 - 2 \ln 5\cr 2x \ln 3 - x \ln 5 &=& -2 \ln 5 - \ln 3\cr x(2 \ln 3 - \ln 5) &=& -(\ln 3 + 2 \ln 5)\cr x &=& \frac{-(\ln 3 + \ln 25)}{2 \ln 3 - \ln 5}\cr &= &\frac{-\ln(75)}{\ln(9/5)}\cr &=& \frac{\ln(75)}{\ln(5/9)}\cr &=& \frac{4.317488}{-0.5877867} \approx -7.34\cr }\]

22. System of equations

Problem: Solve for positive real values \(x\) and \(y\):

1. \(\log_2(x) + \log_2(y) = 5\)
2. \(\log_2(x) - \log_2(y) = 1\)

Solution: Using product and quotient rules of logarithms:

\[\eqalign{ \log_2(x) + \log_2(y) &=& 5\cr \log_2(xy) &=& 5\cr xy = 2^5 &=& 32\cr \cr \hline \cr \log_2(x) - \log_2(y) &=& 1\cr \log_2\left(\frac{x}{y}\right) &=& 1 \cr \frac{x}{y} = 2^1 &=& 2\cr x &=& 2y\cr \cr \hline \cr 2y\cdot y &=& 32\cr 2y^2 &=& 32\cr y^2 &=& 16\cr y &=& 4\cr \cr \hline \cr xy=4x&=& 32\cr x&=& 8\cr }\]

From \(\frac{x}{y} = 2\), substitute \(x = 2y\) into \(xy = 32\):

\[(2y)(y) = 32 \implies 2y^2 = 32 \implies y^2 = 16\]

Since log domains require positive real numbers:

\[y = 4\] ### 23. Solving Equations Quadratic in Form

Problem: Find all real solutions to:

\[e^{2x} - 5e^x + 6 = 0\]

Solution:

\[\eqalign{ e^{2x} - 5e^x + 6 &=& 0\cr \hline u &=& e^x\cr u^2 - 5u + 6 &=& 0\cr (u - 2)(u - 3) &=& 0\cr u &=& (2,3)\cr \hline e^x &=& 2\cr x &=& \ln 2\cr \hline e^x &=& 3\cr x &=& \ln 3\cr \hline x &=& (\ln 2,\ln 3)\cr }\]

24. Extraneous Solutions in Logarithmic Equations.

Problem: Solve for \(x\):

\[\log_6(x + 3) + \log_6(x - 2) = 1\]

Solution:

\[\eqalign{ \log\_6(x + 3) + \log\_6(x - 2) &=& 1\cr \log\_6((x + 3)(x - 2)) &=& 1\cr (x + 3)(x - 2) &=& 6^1 \cr x^2 + x - 6 &=& 6\cr x^2 + x - 12 &=& 0\cr (x + 4)(x - 3) &=& 0 \cr x &=& (-4, 3)\quad but\ reject\ x=-4\cr x &=& 3\cr }\]

25. Change of Base and Algebraic Simplification.

Problem: Simplify the following expression completely without using a calculator:

\[\log_3(8) \cdot \log_2(25) \cdot \log_5(81)\]

Solution:

\[\log_3(8) \cdot \log_2(25) \cdot \log_5(81)\]

\[\log_3(8) = \frac{\ln 8}{\ln 3} = \frac{\ln(2^3)}{\ln 3} = \frac{3 \ln 2}{\ln 3}\]

\[\log_2(25) = \frac{\ln 25}{\ln 2} = \frac{\ln(5^2)}{\ln 2} = \frac{2 \ln 5}{\ln 2}\]

\[\log_5(81) = \frac{\ln 81}{\ln 5} = \frac{\ln(3^4)}{\ln 5} = \frac{4 \ln 3}{\ln 5}\]

\[\left(\frac{3 \ln 2}{\ln 3}\right) \cdot \left(\frac{2 \ln 5}{\ln 2}\right) \cdot \left(\frac{4 \ln 3}{\ln 5}\right)\]

\[= (3 \cdot 2 \cdot 4) \cdot \left(\frac{\ln 2}{\ln 2}\right) \cdot \left(\frac{\ln 5}{\ln 5}\right) \cdot \left(\frac{\ln 3}{\ln 3}\right)\]

\[= 24 \cdot 1 \cdot 1 \cdot 1 = 24\]