Academic Algebra 2. Algebra 1 Review

Transcription

1 Academic Algebra On the following pages you will find a review of the Algebra concepts needed to successfully complete Academic Algebra. Concepts such as fractions, solving equations, inequalities, absolute values, writing equations, graphing linear equations, and linear inequalities are contained in this packet. By reviewing the material covered in Algebra prior to the start of school, will enable us to start directly into the Algebra curriculum. You will be required to complete the packet by the first day of school. We will review the packet the first week of school, and then you will be tested on the information covered in the packet. Students who do not score higher than 8%, will be recommended to take Algebra. I suggest using your Algebra notebook as a resource to assist in completing the packet. If you do not have your notebook, refer to Khan Academy ( ) for video instruction or Google the specific concept. The packet should take about hours to complete. I recommend waiting until August to complete the packet, in order to ensure the concepts are easily recalled for the test. I will have a copy of each page and answer key posted on my web page. If you have any questions during the summer, please contact me via (amoriellok@hasd.us ).

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4 Fractions Show all your work for each problem. No Calculator! Add or Subtract: rewrite with a common denominator and add/subtract Multiplication: cancel and multiple numerator and denominator. Canceling factors: Ex: Multiply or Divide. Division: Take the reciprocal of the second fraction, and multiply \

5 Solving Equations Solve the following linear equations.. 6x 8. x. 4x 8x 4. x. x x 7. 9x 6x 8. x x x x 4 0. x 4. x x. 4 x x 4. 4x 9 x. x 7 8x x 4x 7. x 8 6 x 8. x 6 x 4

6 9. 4 x x x 0. 7x 6 7 x 6. x 6 4 x x. x 8x 4x 8x. x x x 8x 8. 6x 8 9 x x 9x x 4

7 Solving for y Solve the following equations for y. Ex: x y 0 y x 0 0 y x y x. x y. 4x y 7. x y 4. 9x y 6. 4x 8y 6 6. x y x y 0 8. x 4y 7 9. x 9y 0 0. x 7y. x y. 6x y 0 Rewriting Formulas Solve the formula for the indicated variable.. Solve for w: A l w 4. Solve for t: d r t. Solve for r: C r 6. Solve for P: I P rt 7. Solve for b: A b h 8. Solve for h: V r h 4

8 Solving Simple Inequalities -Switch the inequality symbol when you or by a negative number.. x. 6 x 9. 4x x. 7 x 6. 4x 7 7. x x 6 9. x 6 Solving Complex Inequalities ) Distribute ) Combine Like Terms ) Move Variables to one side 4) Move Numbers to the other side 0. x 4. x 8. x 4 6 x 4. x 4 x. 6 x 4x x x 7. x 7 4x 8. 6 x x 6

9 Compound Inequalities: -two simple inequalities joined by and or or. Ex: AND Solve x 8 0 Ex: OR Solve x OR 4x x 8 x 4x 6 x 6 x x 4 Solve the inequality and graph its solution.. x 7. 8 x 6. x x 4 7. x or x 6. 7x 6 or 6x x 0 or x x or x 4 6

10 Absolute Value Equations Ex: x 9 x 9 x 9 Solve the equation. x 4 x 4 x x 7. x 4. x 9. x 4. 4x 7. x x Solve the inequality. Ex: Less Than x 7 AND Ex: Greater Than x 8 OR x 7 x 8 OR x 8 8 x 4 x 6 OR x 0 9 x x OR 0 x 7. x 8. x 4 9. x 7 0. x 8. 4x. 0 6x 6 7

11 Point-Slope Form: y y m( x x ) Ex: m ;,. Substitute m and the point. y y m x ) y x ( x y x. Change to slope-intercept form. y mx b y x 7 Write the equation of the line that passes through the given point with the given slope.. m,,. m,, 4. m,, 4 4. m, 8,. m 0,, 7 6. m,, 7. m 6, 4, 8. m, 7, 9. m 9,, 8 Parallel Lines Ex: y x,,9 -Parallel Lines have the same slope. m y 9 x solve for y. Write the equation of the line that is parallel to the given line and passes through the given point. 0. y x 6,,. y x, 9, 6. y x 6, 8, 7. y x 8,, 4. y 4x,, 7. y x,, 8

12 Perpendicular lines -- opposite reciprocals Ex: y x 6,, -One positive and one negative m -Flip-flop the numbers y x solve for y. Write the equation of the line that is perpendicular to the given line and passes through the given point 6. y x ;, 4 7. y x 9;, 8. y x 7;, 9. y x 6; 8, 4 Two Points: To write an equation given two points: Ex: 4, 8 and, 8 6. Find m. m 4. Select a point.,. Substitute m and the point. y y m( x x ) y x y x 4. Change to slope-intercept form. y mx b y x Write in slope-intercept form the equation of the line that passes through the given points. 0. 0, 8 and,. 7, 9 and,. 4, and,., 7 and, 4. 6, and 0, 4. 4, and 8, 4 9

13 Standard Form: Ax By C -integer coefficients - A is positive Write the equation in standard form. y x 4 x y 4 x y 4 x y Ex:. y x 9. y 6x. 4x 9y x 9 y. y x x 6 y 7. y 6 x 8. y 4 4 x 9. 4 y x 0. y x. y x 7. 7 y x. y x 4. y x

14 Ex: 0, 7 and, m y x y 7 x 0 x y y 7 x x y 7 6 y x 7 6x y Write the equation in slope-intercept form then change it to standard form with integer coefficients.. 6, 8, m 6. 4,, m 7.,, m , 4 and 7, 9. 0, and, 0. 4, 9 and, 6

15 Find b and m. Graph the equation using slope-intercept form (y = mx + b).. y x. y x. y x b = b = b = m = m = m = 4. 4 y x. y x 6 6. y x 4 b= b = b = m = m = m = 7. y 8. y 4 9. y b = b = b = m = m = m =

16 0. y x. y x. y x 4 b = b = b = m = m = m =. y x 4. y x. y x b= b = b = m = m = m = 6. y x 4 7. x 8. x b = b = b = m = m = m =

17 Write the equation in slope-intercept form (y = mx + b). Find b and m, and graph the equation.. x y. x y. x y 6 y = y = y = b = m = b = m = b = m = 4. x y 4. x y 6. x 7y 4 y = y = y = b = m = b = m = b = m = 4

18 7. x y 8. x 4y 0 9. x y 4 y = y = y = b = m = b = m = b = m = 0. x 4y 4. y 4. x y y = y = y = b = m = b = m = b = m =

19 Graphing Linear Inequalities: ) Graph the corresponding equations (Dotted for < > Solid for ) ) Test point Use (0, 0) if possible. ) Shade Solutions Ex: Sketch the graph of x y Corresponding equation: x y y x y x m b Test: (0, 0) true Graph the side with (0, 0) Sketch the graph of each linear inequality.. y. x. y x 4. y x 6

20 . y x 6. y 4x 4 7. y x y x 9. x y 4 0. x y 4 7

21 Answer Key: Page Fractions: ` Page - Solving Equations:. x 4. x 6. x 4. x 9. x 4 6. x 7. x 8. x 4 9. x 0. x. x. x 4. x 7 / 4 4. x. x 6. no solution 7. x 8. x 9 9. x 0. x. x 0. x. x 4 4. x. x 6. x Page 4 Solving for y. y x. y 4x 7. y / x 4. y 9x 6. y / x 6. y /x 7. y 4x 8. y x 7 / 4 9. y /x 0/ 0. y / 7x / 7. y x. y x 0/ Formulas:. w A/ l 4. t d / r. r C / 6. P I / rt 7. b A/ h 8. h V / r Page Simple Inequalities. x 8. x. x 4 4. x 7 / 6. x 6. x 7. x 8. x 9. x 9 0. x /. x. x. x 6 4. x /. x 6. x / 7. x 6 8. x 6. Page 6 Compound Inequalities. x 6. x. 6 x 4. 4 x. x 7 or x 6. x / or x 7. x or x 0 8. x or x 6 Page 7 Absolute Value Equations:. x 9 x. x x 7. x 9 x 4 4. x / x. x 8 x 6/ 6. x 9 x Inequality: 7. x 4 8. x or x / 9. 9 x 0. x or x 0/. 9/ x. x 8 / or x 6 Page 8 Writing Equations of Lines Point-Slope:. y x 7. y x. y x 4. y / 4x. y 7 6. y / x 7. y 6x 8. y x 6 9. y 9x Parallel Lines: 0. y x. y x. y x. y / x 4. y 4x. y / x Page 9 Writing Equations of Lines Perpendicular Lines: 6. y x 6 7. y x 8 8. y / x 9. y / 4x Two Points: 0. y x 8. y 6x. y x 4. y 4x 4. y x 6. y / 4x

22 Page 0 Standard Form. x y 9. 6x y. 4x 9y 4. x y. x y x 4y 6 7. x y x y 9. x y 8 0. x y. x 7y 4. x y 7. 6x 8y 4. 0x y Page Standard Form. x y 4 6. x y x y 8. x y 4 9. x y 0. x y Page - Graphing Slope-Intercept Form. b ; m. b ; m. b ; m 4. b 4; m /. b 6; m / 6. b 0; m / 4 7. b ; m 0 8. b 4; m 0 9. b ; m 0 0. b ; m. b ; m. b 4; m. b ; m / 4. b ; /. b ; m / 6. b 0; m / 4 7. b None; m NoSlope 8. b None; m NoSlope Page 4- Graphing Standard Form. y x ; b ; m. y x ; b ; m. y x 6; b 6; m 4. y x 4; b 4; m

23 . y x ; b ; m 6. y /7x ; b ; m / 7 7. y /x ; b ; m / 8. y / 4x; b 0; m / 4 9. y / x ; b ; m / 0. y / 4x ; b ; m / 4. y ; b ; m 0. y / x / ; b / ; m / Page 6-7 Graphing Linear Inequalities. b ; m 0. b None; m NoSlope. b 0; m 4. b ; m. b ; m 6. b 4; m 4 7. b 4; m / 8. b ; m / 9. b ; m / 0. b 4; m