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UNIT I

EQUATIONS AND INEQUALITIES IN ONE VARIABLE

  • Expressions and Formulas: You will have opportunity to practice order of operations in geometric formulas, and up to the quadratic formula that demands better command of these rules.

  • Real Numbers: In order for you to be able to work in the following sections of this algebra one course, you will have to know the different number sets. All complex numbers are real numbers, but not all real numbers are complex numbers. Or you could say all integers are real numbers, but not all real numbers are integers. Additionally, a review of properties for real numbers is given. The interpretation of the narratives for word problems, and directions further in the book will be easier to understand if you know these concepts.

  • Equations, Including Absolute Value Equations: When you have to solve an equation, you need to decide if you have to add, subtract, multiply, or divide a given term in the equation. You have to do it in both sides of the equation; not to alter the value of the equation. This requires to have a good command of the properties of equality. Solving inequalities is almost the same, except that you deal with multiple solutions for the same variable. Learn to use the properties of equality to solve one variable linear equations and inequalities.

  • Inequalities and Absolute Value Inequalities: If you have to decide what may be the length of a third side in a triangle if they give you two of the sides, then mentally you will try to find it by knowing that this third side needs to be more than the other two sides together, otherwise you can't form the triangle. If they are less, the triangle doesn't "close", if you make it equal then you have two parallel lines tightly together. At the same time, if you choose it too long then the triangle won't "close" for the same reason that when you chose it too short, but now with the new side. This problem is a geometric problem that is solved with three compound inequalities made when you state adding two of the sides more than the third side, and allowing each one of the sides to be the third side in one of the three inequalities. Learn to apply the properties of inequality to solve inequalities and absolute value inequalities.

 

 

 

 

 

 

UNIT II

INTRODUCTION TO LINEAR EQUATIONS

  • Relations, Functions and Linear Equations: We know that in algebra all functions are relations, but not all relations are functions. Why? How may we determine from the graph if we have a relation, or a function? Study the definition of relations and functions. Use of mapping, tables, ordered sets and graph to represent relations and functions. Learn how to apply the vertical test to identify a function. Determine the difference between discrete vs. continues functions. Definition of linear equations and the Standard Form.

  • Learning About Slope: The inclination of a road makes all the difference for a driver of a big truck to decide when driving downhill if he needs to apply the breaks, or slow down his truck with the motor. You might remember, or in your next trip try to see that when the road is too steep; there is a signal with a truck going downhill and a number below and a caption saying: "six degree slope or other equivalent words (or another number), slow down with the motor." and telling for how many miles this is true. They are telling the drivers of big trucks that for each one hundred horizontal feet the road raises vertically six feet. For a full loaded truck downhill this poses the challenge to apply the breaks not to accelerate due to the inertia of the big mass (that is the big weight this implies) on the truck. Problem is that the tendency of the truck is to go faster than safer to go. It needs to be slowed down. If the driver applies the breaks too often they overheat and melt the plastic plugs, gaskets, and seals and then the break liquid leaks out of the break circuit and the truck ends up without a break system to stop it. The solution is to apply a lower gear shift that slows down the truck without applying the breaks. Observe that at the end of those steep slopes you have a safety ramp at the end of a guiding red line in the road full with loose gravel to slow down to a stop any truck for those drivers that didn't learn algebra to read the sign and understanding it. You will learn about slope formula (run vs. rise and two points), falling to the right, horizontal, vertical, negative and positive cases for the slope. Slope of parallel and perpendicular lines.

  • Working With Systems of Linear Equations: Your dad is deciding if getting a new cell phone contract from two competing companies. One charges a fee, and a rate per minute for each call. The other does not charge a fee, but charges a little bit more per minute. His dilemma is to find out after how many minutes the companies will charge exactly the same, knowing that before that minute one charges more than the other and after that point this switches. This problem is solved with a system of linear equations. Each situation may be represented by a straight line in the coordinate plane. Where they intersect is the point your dad wants to find. To solve this and other problems you need to learn about slope intercept form and point-slope form problems that involve a point and slope, two points, etc. Solving systems of two variable linear equations by substitution, linear combinations and graphing. Introduction to special functions (step function, constant function, identity function and absolute value function)
  • Two Variable Linear Inequalities and Absolute Value Inequalities. Like in the section before this one. You will get a straight line in the coordinate plane. The difference is that now this line is the boundary of two areas, where one of them won't include the line itself. One side will have that all points make true the inequality, and at the other all points will make it false. If the inequality is expressed as greater or equal, or as less or equal, then the line will be part of the solution, other wise is not part of it. You will practice solving two variable inequalities by shading above o below the graph to indicate the solution set.

  • Systems of Linear Inequalities and Introduction to Linear Programming and Its Applications. In any game you have legal movements and actions, and you have illegal movements and actions. These second ones are called restrictions. When an engineer in a company has to decide how much rubber, how much wood, how much oil, etc... for the construction of a skateboard that needs rubber for the wheels, wood for the board, and oil to lubricate the bearings of the wheels. He needs to be careful when requesting the quantities of each, so that he doesn't end up with too much, or too many of one, and two little, or too few of other components. That is done setting up a system of inequalities in the coordinate plane. They enclose one area inside a polygon. The points of interest are the vertices of the polygon. With those coordinates they use a function that is evaluated and helps to decide the maximum or minimum of all the existent coordinates in the points of the polygon. The inequalities are the "restrictions" he needs to take into account. The function to evaluate is the optimization function to be maximized or minimized. Learn how to solve systems of two variable linear inequalities. This concept it is extended to introduce concepts as solution region, feasible region, and maximum and minimum of the optimization function. Solution of linear programming problems.

 

 

 

 

 

 

UNIT III

MANIPULATING POLYNOMIALS

  • Polynomials: Learn to apply the properties of exponents to simplify polynomial expressions.
  • Adding polynomials: The same way that when you add or subtract in arithmetic, you lineup ones, tens, hundreds, thousands, etc. In arithmetic of polynomials you may lineup constant terms, linear terms, quadratic terms, etc. to perform the addition or subtraction in vertical format. Learn to add and subtract polynomials. Horizontal and vertical formats.
  • Manipulate Radicals and Rational Exponents. A radical may be expressed as a quantity to a fractional power, and vice versa. The advantage is that if you have to do operations with radicals you change them to form where they as expressed as the power of a quantity and then you apply the rules of exponents for addition, subtraction, multiplication, and division to perform the necessary operations. Learn to simplify expressions that contain radicals and rational exponents.
  • Radical Equations and Inequalities. Radical equations pose a challenge. You need to determine for what set of values the radicand inside the radical yields real solutions. The reason is that otherwise you end up dealing with imaginary solutions that are not part of the algebra 1 curriculum. Then you need to review inequalities to be able to perform this check. Also, the radical expression itself may be one of the terms of the equation or inequality and this needs a similar check. Study how to solve radical equations and inequalities.

 

 

 

 

 

 

 

UNIT IV

QUADRATIC FUNCTIONS

  • Quadratic Functions and Its Roots. In algebra one you don't deal with the general equation for conics that may have as geometric space (graph) a circle, a parabola, a hyperbola, or a ellipse. You just deal with a second degree polynomial in the right side of the equation that has as geometric space, the parabola. Nevertheless, to find the zeros, roots, solutions, or x-intercepts; you need to solve quadratic functions using several methods. For example: By factoring, completing the square and by use of the Quadratic Formula. This presentation dwells in each one of them with morbid attention to detail in order to help you not to miss, what you might have missed before.

  • Discriminant, Product and Sum of Roots. If you graph a parabola just by selecting a feasible domain to display it with vertex and whether it opens up or down, right or left. Then you will discover that sometimes it intersects the x-axis in two points, sometimes in only one, and sometimes it does not intersect at all (vertical parabolas). You are going to have the ability at the end of this lesson to determine the number of roots of a quadratic equation by looking at the radicand of the quadratic formula, and evaluating it to determine if is more, equal, or less than zero. This in turn will tell you if the parabola will intersect in two points, one, or none the x-axis. Additionally, you will practice how to verify the solution with the product and sum of its roots.