Compound inequalities often have three parts and can be rewritten as two independent inequalities.See Example 3, Example 4, Example 5, and Example 6. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality. Solving inequalities is similar to solving equations.Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well. Interval notation is a method to indicate the solution set to an inequality.See Example 11.ΔΆ.7 Linear Inequalities and Absolute Value Inequalities Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form.We may also see a binomial in place of the single variable. Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the second term and the third term is a constant.To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value.We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index.Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping.To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. Rational exponents can be rewritten several ways depending on what is most convenient for the problem.Solving for the length of one side of a right triangle requires solving a quadratic equation. The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields.The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each.A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation.Completing the square is a method of solving quadratic equations when the equation cannot be factored.The solution will yield a positive and negative solution. We isolate the squared term and take the square root of both sides of the equation. Another method for solving quadratics is the square root property.Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method.The zero-product property is then used to find solutions. Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the equation is a difference of squares.Perpendicular lines have slopes that are negative reciprocals of each other unless one is horizontal and the other is vertical.Parallel lines have the same slope and different y-intercepts.Vertical lines have an undefined slope (zero in the denominator), and are defined as x = c, x = c, where c is a constant.Horizontal lines have a slope of zero and are defined as y = c, y = c, where c is a constant.The standard form of a line has no fractions.Find the slope and use the point-slope formula. We can also find the equation of a line given two points.We can find the equation of a line given the slope and a point.We can identify the slope and y-intercept of an equation in slope-intercept form.Given two points, we can find the slope of a line using the slope formula.See Example 5 and Example 6 and Example 7. All solutions to a rational equation should be verified within the original equation to avoid an undefined term, or zero in the denominator.We use the LCD to clear the fractions from an equation. A rational expression is a quotient of two polynomials.We can solve linear equations in one variable in the form a x + b = 0 a x + b = 0 using standard algebraic properties.The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the x-coordinates and the sum of the y-coordinates of the endpoints by 2.The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment.These are the points where the graph crosses the axes. Finding the x- and y-intercepts can define the graph of a line.Equations usually have to be entered in the form y=_. Using a graphing calculator or a computer program makes graphing equations faster and more accurate.An equation can be graphed in the plane by creating a table of values and plotting points.We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as displacement from the x-axis and displacement from the y-axis.2.1 The Rectangular Coordinate Systems and Graphs
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