H-AT THE INTEGERS UNIT POLYNOMIALS AND THE NUMBER LINE (DAY ) Warm-Up: Find the solution set of the following inequality: 0 5 < 40 INEQUALITY SOLUTION SETS Solve the inequality equation Graph the solution on a, using correct endpoints and determine shading by using a test point. o After solution is graphed, there are different ways to notate the set of answers for an inequality equation, use one of the following notations. Use round parenthesis/square brackets and a comma between the lower limit and the upper limit numbers. Use inequality symbols (<, >,, ) and the variable. SYMBOL in problem < or > Endpoints on number line SET NOTATION -5 < < < -5 or > INTERVAL NOTATION MEANING The lower and upper limit numbers are not included in the solution set. or -5-5 or The lower and upper limit numbers are included in the solution set. Shading that doesn t stop is headed towards or. ALWAYS! Epress the following solution set in interval notation and set notation. ) ) - -
) 4) -4.5 4.5 5) 6) 0-0 Adding/Subtracting Polynomials: Combine two or more terms that have the same power of the same variable into a single term. Simplify the following: 7) (a b ab + 5) + (a b + ab ) 8) (8 ) ( 0 + ) 9) ( - + ) + ( + - 7) 0) ( 4 + 5 - ) ( - + 4 ) ) ( - )( + ) ) ( 5-7) ) What is the difference of 5 9 - - 6 + subtracted from 5 + 6 4 - -?
ABSOLUTE VALUE EQUATIONS (DAY ) Absolute Value: is the away from on the number line. ) 5 ) = ) - 6 = 4 ) - 5 = If you are given = 4, that means that the could have started out as or. Find the possible values for a: a = 5 If 7, then can equal: STEPS FOR SOLVING ABSOLUTE VALUE EQUATIONS 4 Procedure 55 If not done already, isolate the absolute value. - Remove the absolute value symbol. - Create equations: ( st ) equal to a positive result ( nd ) equal to a negative result Solve each equation. Check the solutions in the original. Write answer as a solution set.
Solve each of the following equations for the given variable. ) 4a 7 ) 6 5 ) 5 4 8 4) 6 8 4
ABSOLUTE VALUE INEQUALITIES (DAY ) Warm-Up: Subtract + 8 4 from 7 5 + STEPS FOR SOLVING ABSOLUTE VALUE INEQUALITIES Procedure 5 Isolate the absolute value, if necessary. Change inequality sign to an equal sign. TO FIND THE ENDPOINTS Write equations: ( st ) Drop the absolute value but keep everything else the same ( nd ) Drop the absolute value; make the result negative Solve each inequality for. Writing Inequality Solution: Graph ENDPOINTS on a number line. Choose correct open/closed circles, determine shading by using a test point. (shade where test point is TRUE) Write a solution set using either interval or set notation. Solve the following Inequality Equations. ) 4 6 ) + - 8 > 5
5 ) 4) 5 5) What is the solution set of the equation 5? () {-,} () {-} () {} (4) {,-} 6) The solution set of which inequality is represented by the accompanying graph? () 7 () 7 () 7 (4) 7 7) A depth finder shows that the water in a certain place is 60 feet deep. The difference between d, the actual depth of the water, and the readings is d 60 and must be less than or equal to 0.05d. Find the minimum and maimum values of d, to the nearest tenth of a foot. 6
FACTORING STEPS D: Greatest Common Factor FACTORING POLYNOMIALS (DAY 4) D: Difference Between Two Perfect Squares D: Quadratic Trinomial (AC method) D4: Factor by grouping Factor the following: ) ( 5) ( 5) ) - 6 + 9 ) 5yz 5 4) 5 0 9 6 5) - 9 6) 49 y 7) ( ) 5( ) 8) a 4 9) 5 4 5 0) 8 ) b 6 5a 4 ) a b ab c 00 ) 9 8 4) + 4 5) 6 4 7
Recall Factoring Steps: MORE FACTORING (DAY 5) Factor each of the following. ) + 5 5 4) 6 + 7 + ) + 5 5) 6 + 5 6 ) 7-0 + 6) 5-6 7) - + + 0 8) 5 + 4-8
SOLVING QUADRATIC EQUATIONS (DAY 6 AND 7) ) Solve the following for : + 4 = 0 Graph the parabola in your calculator, and find your answers on the graph. These numbers are also called: Factoring a quadratic equation gives you: Solving a quadratic equation gives you: Find the roots of the following equations & confirm in your calculator. ) 6 = 0 ) 6 = 0 4) = 6 + 5) Factoring and solving are different because: 6) If + is a factor of a polynomial, then what would a root of that polynomial be? 7) Write a quadratic equation that has roots of 7 and 4. 8) Write a quadratic equation that has roots of and. 9
SOLVING QUADRATIC INEQUALITIES Solve: 6 < - Procedure ) Rewrite the inequality in standard form. **Remember it will be easier if the squared term is positive before you factor. ) Factor like a regular quadratic equation. ) Set factors equal to zero and solve. 4) Plot roots (endpoints) on the number line with correct open/closed circle. 5) Pick a test point to determine where to shade. (shade where test point is TRUE) 6) Write solution set in either interval or set notation Solve the following quadratic inequalities: ) + 4 < 0 ) -5 < - 0
) d 9 0 4) + 0 5) A rectangular floor can be covered completely with tiles that each measure one square foot. The length of the floor is foot longer than the width and the area is greater than 56 square feet. What are the smallest possible integral dimensions of the floor?