Algebra: Equations and Inequalities

Order of Operations

• To simplify expressions, use the order of operations
• The order to follow is Parentheses, Exponents, Multiplication and Division, Addition and Subtraction, commonly remembered as PEMDAS

Solving Equations

• To solve the equation 5(x - 4) + 5x = 10(2 - x), the value is x = 2
• To solve the equation 5(x – 3) – 6x =10(3−x), the value is x = 5
• To solve the equation 1/2x+1/4 = 3/4x-1/6, the value is x = 5/3
• To solve the equation 1/3x-1/5 = 1/5x+1/3, the value is x = 4

Translating algebraic equations

• "Twice the difference of a number and three is equal to eight" translates to 2(n − 3) = 8; n = 7
• "Eight less than the product of three and a number is twenty-two" translates to 3n-8 = 22; n = 10
• "Four times a number added to eight is equivalent to the opposite of four" translates to 4n + 8 = −4; n = −3

Solving Inequalities

• To solve the inequality 2(x-5)+5 ≤ -21, the answer is x ≤ -8, which is graphed on a number line with a closed circle at -8 and an arrow pointing to the left
• To solve the inequality -3(4x-6) > −10x, the answer is x<9, which is graphed on a number line with an open circle at 9 and an arrow pointing to the left

Graphing Linear Equations

• The graph of 4x-2y = 8 is an increasing line passing through (0,-4) and (2,0)
• The graph of 2x + 4y = 8 is a decreasing line passing through (0,2) and (4,0)

Finding Slope

• For the points (0, -1) and (3, 2), the slope m = 1, which is an increasing line due to positive slope
• For the points (-2, 7) and (-1, 2), the slope is m = -5, which is a decreasing line due to negative slope
• For the points (9, -3) and (5, -3), the slope is m = 0, which is a horizontal line due to zero slope

Linear Equations

• Rewriting 3x+5y=8 in slope-intercept form gives y=-3/5x+8/5, with a slope of -3/5 and a y-intercept of (0, 8/5)
• Rewriting 2x-4y=12 in slope-intercept form gives y=1/2x-3, with a slope of 1/2 and a y-intercept of (0,-3)
• The equation of the line with a slope of m = -2 and y-intercept of (0,3) is y = -2x+3
• The equation of the line with a slope of m = 3 and y-intercept of (0,-2) is y = 3x-2
• The equation of the line with a slope of m = 4 passing through point (5,3) is y=4x-17
• The equation of the line with a slope of m = -6 passing through point (−1,2) is y=-6x-4

Systems of Equations

• Solving the system 7x-4y = 4 and 5x + y = 26 gives (4,6)
• Solving the system 3x – 5y = -17 and y=-15-4x gives (-4,1)

Word Problems (Systems of Equations)

• If a restaurant manager buys 50 lb of sausage and 80 lb of hamburger for $300, and 100 lb of sausage and 120 lb of hamburger for $480, the equations are 50x + 80y = 300 and 100x +120y = 480
  • The cost of hamburger is $3.00 per pound and the cost of sausage is $1.20 per pound.
• If David has 39 bills in his wallet worth $330, all fives and tens, the equations are x + y = 39 and 5x+10y = 330
  • David has 12 ten dollar bills and 27 five dollar bills.

Simplifying Expressions

• (9n³ + 5nm² + nm − 11) – (−2n³ – nm + 15) simplifies to 11n³ + 5nm² + 2nm – 26
• (−9u³ - 5uv² + vu − 11) + (u³ − 13vu + 15) simplifies to – 8u³ – 5uv² – 12uv + 4

Multiplying and Simplifying

• (11y - 9)(15y + 3) simplifies to 165y² - 102y – 27
• (6m – 2)² simplifies to 36m² - 24m + 4
• (3x - 7y)(3x + 7y) simplifies to 9x²-49y²
• (5x + 4)² simplifies to 25x² + 40x+16

Simplifying Expressions with Exponents

• (-3/7xy²z⁴)³ simplifies to -27/343x³y⁶z¹²
• 2(-6/11a³bc⁴)² simplifies to 72/121a⁶b²c⁸
• 4x⁷yz³/(-6x²z⁵) simplifies to -2x⁵y/3z²
• -10x⁵yz²/(25x²z⁸) simplifies to -2x³y/5z⁶

Surface Area of a Box

• The surface area of a box is described by the polynomial S = 2LW + 2LH + 2WH.
• A box with length 8 inches, width 6.5 inches, and height 4 inches has a surface area of 220 square inches.

Height of a Baseball

• The height of a baseball after t seconds is modeled by h = −16t² + 100t + 4.
• After four seconds, the height of the baseball is 148 feet.

Factoring Completely

• 2x² + 3x + 4xy + 6y factors to (2x + 3)(x + 2y)
• 5ab² – 20ab – 105a factors to 5a(b−7)(b + 3)
• 4x² - 49y² factors to (2x-7y)(2x + 7y)
• 2x³-14x² + 24x factors to 2x(x-3)(x-4)

Solving Equations by Factoring

• m² - m = 6 has solutions m=-2 and m=3
• 2x²-9x+4=0 has solutions x=1/2 and x=4
• 5x²-14x-3 = 0 has solutions x = 3 and x = -1/5

Word Problems

• A building's length is twice its width, and the floor area is 288 square feet, modeled by 2w(w) = 288, meaning the width is 12 feet and the length is 24 feet.
• A rectangle's length is 7 meters more than its width with an area of 78 square meters, modeled by w(w+7) = 78, which means the width is 6 meters and the length is 13 meters.

Factoring and Simplifying

• (x² - 5x) / (x² - 7x + 10) simplifies to x/(x-2)
• (x² - 9) / (x² + 5x + 6) simplifies to (x-3)/(x+2)

Performing Operations and Simplifying

• (y² - 6y + 5)/(y² - 1) * (y - 1)/(y² - 10y + 25) simplifies to (y-1) / ((y +1)(y - 5))
• (x² - 2x - 24)/(x² - 16) ÷ (x² - x - 30)/(x² + 10x + 25) simplifies to (x+5)/(x-4)

Performing Operations and Simplifying

• (x² + 2)/(x + 1) + (4 - x²)/(x + 1) simplifies to 6/(x+1)
• y²/(y² + 3y) - 9/(y² + 3y) simplifies to (y-3)/y

Solving Equations

• 3/x = 5/(x-8) has the solution x=-12
• 2/(3(x-2)) = -1/(-2(3-x)) has the solution x = 18/7

Simplifying Radical Expressions

• √5x³ * √20x simplifies to 10x²
• √2x⁴ * √32x⁸ simplifies to 8x⁶
• √48a⁷ / √3a simplifies to 4a³
• √72x³ / √2x simplifies to 6x

Solving Radical Equations

• √(2x - 1) = 6 has the solution x = 37/2
• √(x - 3) + 5 = 11 has the solution x = 39
• √(x + 1) - 4 = 3 has the solution x = 48
• √(x + 3) + 2 = 1 has no solution

Quadratic Formula

• The solutions to 6x² - 3x - 4 = 0 are given by x = (3 ± √105) / 12
• The solutions to 4x² - 4x - 1 = 0 are given by x = (1 ± √2) / 2

Quadratic Equations

• For y = x² - 4x + 3: the vertex is (2,-1), the y-intercept is (0,3), and the x-intercepts are (1,0) and (3,0)
• For y = x² + 2x - 8: the vertex is (-1,-9), the y-intercept is (0,-8), and the x-intercepts are (-4,0) and (2,0)