• Test (0,0): \( 0 > 2 \) is FALSE → Shade above!
x y
Sample Solution Point: ( 2 , 2 ) or ( 0 , 4 )
7. Standard Form Inequality Key
\( 2x - 3y \le 6 \)
• Intercepts: \( x \)-int: \( (3, 0) \) • \( y \)-int: \( (0, -2) \)
• Slope-Int: \( -3y \le -2x + 6 \implies \mathbf{y \ge \frac{2}{3}x - 2} \)
• Boundary: Solid Line • Test (0,0): \( 0 \le 6 \) (True!)
x y
Sample Solution Point: ( 0 , 0 ) or ( -1 , 2 )
8. System of Linear Inequalities Key
\( \begin{cases} y \ge \frac{1}{2}x - 2 \\ 2x + y < 3 \end{cases} \)
Line 1: \( y = \frac{1}{2}x - 2 \)
• Solid line; \( y \)-int \( (0, -2) \), slope \( \frac{1}{2} \)
• Test (0,0): \( 0 \ge -2 \) (True → shade ABOVE)
Line 2: \( y < -2x + 3 \)
• Dashed line; \( y \)-int \( (0, 3) \), slope \( -2 \)
• Test (0,0): \( 0 < 3 \) (True → shade BELOW)
Target Solution Region: Intersection area contains \( (0, 0) \).
x y S
Part 4 • Solving Systems of Linear Equations • Answer Key Page 3 Key
9. Solve by Graphing Key
\( \begin{cases} y = 2x - 3 \\ y = -\frac{1}{2}x + 2 \end{cases} \)
x y (2, 1)
Check: \( 1 = 2(2) - 3 = 1 \) • \( 1 = -\frac{1}{2}(2) + 2 = 1 \) ✓
Solution: ( 2 , 1 )
10. Solve by Substitution Key
\( \begin{cases} x = 3y - 1 \\ 2x + 4y = 18 \end{cases} \)
Step 1: Substitute \( x = 3y - 1 \) into Eq 2:
\( 2(3y - 1) + 4y = 18 \)
\( 6y - 2 + 4y = 18 \)
\( 10y - 2 = 18 \implies 10y = 20 \implies \mathbf{y = 2} \)
Step 2: Back-substitute to find \( x \):
\( x = 3(2) - 1 = 6 - 1 = \mathbf{5} \)
Step 3: Check in Eq 2:
\( 2(5) + 4(2) = 10 + 8 = 18 \) ✓
Solution: ( 5 , 2 )
11. Solve by Elimination Key
\( \begin{cases} 3x + 2y = 7 \\ 5x - 4y = 26 \end{cases} \)
Step 1: Multiply Eq 1 by 2 to match \( y \):
\( 2(3x + 2y = 7) \implies \mathbf{6x + 4y = 14} \)
Step 2: Add the two equations:
\( (6x + 4y) + (5x - 4y) = 14 + 26 \)
\( 11x = 40 \implies \mathbf{x = \frac{40}{11}} \)
Step 3: Solve for \( y \):
\( 3\left(\frac{40}{11}\right) + 2y = 7 \implies \frac{120}{11} + 2y = \frac{77}{11} \)
\( 2y = -\frac{43}{11} \implies \mathbf{y = -\frac{43}{22}} \)
Solution: \( \left(\frac{40}{11}, -\frac{43}{22}\right) \)
12. Perimeter Application Key
Variables: Let \( L = \) length, \( W = \) width
Equation 1: \( L = 2W + 6 \)
Equation 2: \( 2L + 2W = 60 \) (or \( L + W = 30 \))
Step 1: Substitute \( L = 2W + 6 \) into Eq 2:
\( 2(2W + 6) + 2W = 60 \)
\( 4W + 12 + 2W = 60 \implies 6W = 48 \implies \mathbf{W = 8\text{ cm}} \)
Step 2: Solve for \( L \):
\( L = 2(8) + 6 = 16 + 6 = \mathbf{22\text{ cm}} \)
Step 3: Verification Check:
Perimeter: \( 2(22) + 2(8) = 44 + 16 = 60\text{ cm} \) ✓
Length \( L = 22 \) cm Width \( W = 8 \) cm
Part 5 • Exponents, Radicals & Exponential Equations • Answer Key Page 4 Key
Part 6 • Simplifying with Properties of Exponents Use only positive exponents
13. Product & Power Rules
\( (-3x^4 y^2)^3 \cdot (2x^2 y^5) \)
\( = (-27 x^{12} y^6) \cdot (2x^2 y^5) \)
\( = (-27 \cdot 2) \cdot x^{12+2} \cdot y^{6+5} \)
Answer: \( -54x^{14}y^{11} \)
14. Quotient & Negative Rules
\( \frac{24a^5 b^{-3} c^0}{16a^{-2} b^4 c^6} \)
\( c^0 = 1 \); \( \frac{24}{16} = \frac{3}{2} \)
\( a^{5 - (-2)} = a^7 \); \( b^{-3 - 4} = b^{-7} = \frac{1}{b^7} \)
Answer: \( \frac{3a^7}{2b^7 c^6} \)
15. Power of a Quotient
\( \left( \frac{2m^3 n^{-4}}{m^{-2} n^2} \right)^{-2} \)
Inside: \( 2m^{3 - (-2)} n^{-4 - 2} = \frac{2m^5}{n^6} \)
Apply \( -2 \): \( \left(\frac{n^6}{2m^5}\right)^2 \)
Answer: \( \frac{n^{12}}{4m^{10}} \)
16. Mixed Operations
\( \frac{(4p^{-3} q^5)^2}{8p^2 q^{-1}} \)
Numerator: \( 16p^{-6}q^{10} \)
\( \frac{16}{8} = 2 \); \( p^{-6 - 2} = p^{-8} \); \( q^{10 - (-1)} = q^{11} \)
Answer: \( \frac{2q^{11}}{p^8} \)
Part 7 • Converting Rational Exponents & Radicals \( a^{m/n} = \sqrt[n]{a^m} \)
Radical Form:
17. \( x^{3/5} \) \( \sqrt[5]{x^3} \) or \( (\sqrt[5]{x})^3 \)
18. \( (5y)^{2/3} \) \( \sqrt[3]{(5y)^2} = \sqrt[3]{25y^2} \)
Rational Exponent Form:
19. \( \sqrt[4]{k^7} \) \( k^{7/4} \)
20. \( \frac{1}{\sqrt[3]{w^5}} \) \( \frac{1}{w^{5/3}} = w^{-5/3} \)
Part 8 • Evaluating with Rational Exponents Evaluate without a calculator
\( 64^{2/3} \)
\( (\sqrt[3]{64})^2 = 4^2 \)
Value: 16
\( 81^{-3/4} \)
\( \frac{1}{(\sqrt[4]{81})^3} = \frac{1}{3^3} \)
Value: \( \frac{1}{27} \)
\( (-32)^{3/5} \)
\( (\sqrt[5]{-32})^3 = (-2)^3 \)
Value: -8
\( \left(\frac{16}{49}\right)^{-1/2} \)
\( \sqrt{\frac{49}{16}} = \frac{7}{4} \)
Value: \( \frac{7}{4} \)
Part 9 • Solving Exponential Equations Equate like bases: \( b^u = b^v \implies u = v \)
25. Single Base Match
\( 2^{3x - 1} = 32 \)
\( 32 = 2^5 \)
\( 3x - 1 = 5 \implies 3x = 6 \)
\( x = 2 \)
26. Rewrite Both Bases
\( 9^{2x + 1} = 27^{x - 2} \)
\( (3^2)^{2x+1} = (3^3)^{x-2} \)
\( 4x + 2 = 3x - 6 \)
\( x = -8 \)
27. Negative Exponent Base
\( 5^{4x - 3} = \frac{1}{125} \)
\( \frac{1}{125} = 5^{-3} \)
\( 4x - 3 = -3 \implies 4x = 0 \)
\( x = 0 \)
