5.7 Quadratic Inequalities
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This document discusses how to graph and solve quadratic inequalities in one and two variables. It provides examples of graphing quadratic inequalities by drawing the parabola, choosing a test point, and shading the appropriate region. Systems of quadratic inequalities can be graphed on the same grid, with the shared shaded region representing the solutions. Quadratic inequalities in one variable can also be solved by graphing the parabola and identifying the x-values where it lies above or below the x-axis, or algebraically by solving the inequality as an equation and testing values between the solutions.
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Graphs of quadratic inequalities will be parabolas with solid or dotted lines and shaded regions above or below the parabola. To graph: 1) sketch the parabola y=ax^2+bx+c, 2) choose a test point and see if it satisfies the inequality, 3) shade the appropriate region based on the test point. For example, to graph y ≤ x^2+6x-4: the parabola opens up with a solid line and vertex at (-3,-13), the test point (0,0) satisfies the inequality so shade below the parabola.
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Here are the steps to solve each inequality and graph the solution:
1) m + 14 < 4
-14 -14
m < -10
Graph: m < -10
2) -7 > y-1
+1 +1
-6 > y
y < -6
Graph: y < -6
3) (-3)k < 10(-3)
-3
k > -30
Graph: k > -30
4) 2x + 5 ≤ x + 1
-x -x
x + 5 ≤ 1
-5 -5
x ≤ -4
Graph: x ≤ -4
The angles family
The document introduces the Angle family, consisting of Mr. Obtuse Angle, Mrs. Right Angle, Baby Acute Angle, and their pet Reflex Angle. Mr. Obtuse Angle is always inside and measures between 90 and 180 degrees. Mrs. Right Angle always measures 90 degrees and believes she is always right. Baby Acute Angle is the smallest family member and loves hugs. Reflex Angle is their pet dog who is currently 180 degrees but can grow to 360 degrees, making him the biggest angle.
Lines and angles
For those who need help in PPT's for Lines and Angles and want to get good results.
Visit my website :- http://www.soumyamodakbed.blogspot.in/ for more information.
Chapter 3. linear equation and linear equalities in one variables
Here are the steps to solve this inequality problem:
1) Write an expression for the perimeter in terms of x
2) Set the perimeter expression ≤ 40
3) Isolate x by undoing the operations
4) Write the solution set
The solution is 0 ≤ x ≤ 7
Angles ppt
1) The document defines an angle as being formed by two rays with a common endpoint called the vertex. Angles can have points in their interior, exterior, or on the angle.
2) There are three main ways to name an angle: using three points with the vertex in the middle, using just the vertex point when it is the only angle with that vertex, or using a number within the angle.
3) There are four types of angles: acute, right, obtuse, and straight. Angles are measured in degrees with a full circle being 360 degrees.
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Chapter 3. linear equation and linear equalities in one variables
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Linear ineqns. and statistics
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January 11
1) The document discusses graphing linear inequalities on a number line and coordinate plane. It provides examples of solving inequalities for y and graphing the corresponding boundary lines, shading the appropriate regions.
2) Methods for graphing inequalities include solving for y, graphing the boundary line, and shading the correct region based on whether the inequality is <, ≤, >, or ≥.
3) An example problem models an inequality describing the maximum number of nickels and dimes that can be had with less than $5.00, graphing the solution on the n-d plane.
Hprec2 5
This document provides an overview of solving inequalities, including:
1) The principles for solving inequalities are similar to equations but the direction of the inequality changes when multiplying or dividing by a negative number.
2) Interval notation is used to represent the solutions of inequalities on the number line.
3) To solve inequalities, determine the zeros of the function and where the graph is above or below the x-axis to identify the intervals of solutions.
Mc ty-cubicequations-2009-1
1) Cubic equations can have one real root or three real roots. They always have at least one real root.
2) Cubic equations can be solved by using synthetic division to reduce them to a quadratic equation if one root is known, or by spotting factors.
3) Graphs of cubic equations cross the x-axis at least once, showing they have at least one real root, and can help locate approximate solutions.
Mc ty-cubicequations-2009-1
1) Cubic equations can have one real root or three real roots. They always have at least one real root.
2) Cubic equations can be solved by using synthetic division to reduce them to a quadratic equation if one root is known, or by spotting factors.
3) Graphs of cubic equations cross the x-axis at least once, showing they have at least one real root, and can help locate approximate solutions.
Lecture 07 graphing linear equations
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March 18
Okay, let's break this down step-by-step:
* 90% of 90 girls showed up on time
* 90% is 0.9
* 0.9 * 90 is 81 girls on time
* The total number of girls is 90
* So the number of girls late is 90 - 81 = 9
* 80% of 110 boys showed up on time
* 80% is 0.8
* 0.8 * 110 is 88 boys on time
* The total number of boys is 110
* So the number of boys late is 110 - 88 = 22
* The total number of children late is the girls (9) + the boys (22)
* So the total number
Jan. 13 polynonial sketching
The document discusses how to sketch graphs of polynomial functions. It explains that for odd degree polynomials, the graph behavior is opposite on the left and right sides, while for even degree polynomials the behavior is similar on both sides. It provides examples of how the graphs of cubic, quartic and quintic polynomials will appear. It outlines steps for sketching a polynomial graph which include finding the y-intercept, roots, sign of the function over intervals, and then sketching the graph. An example problem is given to factor a polynomial and sketch its graph.
8 inequalities and sign charts x
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Quadratic functions my maths presentation
1. The document discusses key concepts related to quadratic equations including factorizing, solving, graphing, finding roots and axis of symmetry, locating maximum/minimum values, and using graphs to solve related equations and inequalities.
2. The key steps provided to factorize a quadratic equation are to find the product and factors of the coefficients that add up to the middle term, and then pair the factors.
3. To solve a quadratic equation, take all terms to one side, factorize, set each factor equal to 0 and solve for x to find the two roots.
Quadratic Equations
Quadratic equations take the form ax^2 + bx + c = 0. This document discusses four methods for solving quadratic equations: factorizing, completing the square, using the quadratic formula, and graphing. It provides examples of solving quadratic equations with each method and emphasizes that practice is needed to master the techniques.
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Math Algebra
The document provides an overview of key algebra concepts covered on the ACT exam, including expressions, equations, inequalities, functions, and quadratic equations. It discusses how to simplify and combine like terms in expressions, solve different types of equations (linear, quadratic, absolute value), factor expressions using techniques like greatest common factor and difference of squares, work with fractions and systems of equations. Example problems and step-by-step explanations are provided for many of the concepts. The document is intended as a review of essential algebra skills and strategies for tackling related questions that may appear on the ACT.
chapter1_part2.pdf
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2. Key concepts explained include slope as rise over run, the different forms of writing a linear equation, finding the x- and y-intercepts, and using two points to write the equation of a line in slope-intercept form.
3. Examples are provided to demonstrate how to graph lines based on their equations in different forms, find intercepts, write equations from two points, and solve systems of linear equations.
PPT 6.8 Graphing Linear Inequalities.ppt
1. Solve the inequality for y: y > 2x - 3.5
2. Graph the boundary line y = 2x - 3.5 as a solid line since the inequality symbol is <
3. Test a point, such as the origin (0,0), which satisfies the inequality so shade below the boundary line where the solutions lie.
PPT 6.8 Graphing Linear Inequalities.ppt
1) Solve the inequality for y to get y > 2x - 3.5 and graph the boundary line y = 2x - 3.5 as a solid line. 2) Since the inequality symbol is <, shade below the boundary line where the solutions lie. 3) Check a point not on the line, like (0,0), to verify it does not satisfy the inequality and confirm the shading is below the boundary line.
Maths
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Solving quadratics by graphing
This document discusses how to solve quadratic equations by graphing. It explains that a quadratic equation is of the form y = ax^2 + bx + c, with a being the quadratic term, b the linear term, and c the constant. The number of real solutions is at most two: one solution, two solutions, or no solutions. Solutions are found by setting the equation equal to 0 and finding the x-intercepts of the graph. The graph of a quadratic is a parabola with the roots as the x-intercepts and the vertex as the maximum or minimum point. Quadratic equations can be graphed by making a table of x and y values to plot the parabola.
System of linear inequalities
Here is the system of inequalities for the cross-country team problem:
Let x = number of water bottles sold to students
Let y = number of water bottles sold to others
x + y ≤ 100 (they have 100 bottles total)
3x + 5y ≥ 400 (they need at least $400)
Graph the regions defined by these inequalities and the overlapping region is the solution.
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6 5 parabolas
This document provides an overview of parabolas including their key properties and examples of how to graph and write equations of parabolas. It defines that every point on a parabola is equidistant from the focus and directrix, with the directrix perpendicular to the line of symmetry and the vertex halfway between them. Examples are given of finding the focus, directrix, and graphing parabolas given these features as well as writing the equation of a parabola given its directrix.
4.3 finding missing angles
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R.3 solving 1 var inequalities
This document provides instructions for solving inequalities and graphing their solutions. It notes that when multiplying or dividing an inequality by a negative number, the inequality symbol is reversed. Open circles are used for < and >, while closed circles are used for ≤ and ≥. Examples are provided of solving various inequalities algebraically and graphing the resulting solution sets on a number line.
R.2 solving multi step equations
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10 3 double and half-angle formulas
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2) It derives formulas that relate trig functions of double and half angles to trig functions of the original angle, such as sin(2x) = 2sin(x)cos(x) and sin(x/2) = ±√(1-cos(x))/2.
3) Examples are provided to demonstrate applying these formulas to simplify trigonometric expressions and derive new identities.
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3) Examples are provided to demonstrate applying these formulas to find the exact value of tangents of sums/differences of known angles.
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5.7 Quadratic Inequalities
- 1. 5.7 Graphing and Solving Quadratic Inequalities
- 2. Quadratic Inequalities in Two Variables Four types: To graph: 1. Draw the parabola. >,< : dashed line , : solid line 2. Choose a point from the inside and plug it in 3. If true, shade inside. If false, shade outside.
- 3. Example: Graph y > x2 – 2x – 3 Remember, find the vertex!
- 4. Your Turn! Graph y 2x2 - 5x – 3
- 5. Graphing a System Graph both inequalities on the same grid. Solutions are where the shading overlaps. Example: Graph
- 6. Quadratic Inequalities in One Variable Can be solved using a graph. To solve ax2 + bx + c < 0 (or 0), graph the parabola and identify on which x-values the graph lies below the x-axis. To solve ax2 + bx + c > 0 (or 0), identify on which x-values the graph lies above the x-axis. Find the intercepts by solving for x. (Remember: solutions = zeros = x-intercepts)
- 7. Solving by Graphing Solve x2 – 6x + 5 < 0 Solution: 1<x<5
- 8. Example: Solve 2x2 + 3x – 3 0
- 9. Your Turn! Solve -x2 – 9x + 36 > 0
- 10. Solving Algebraically 1. Write as an equation and solve. 2. Plot the solutions (called critical x-values) on a number line. 3. Test an x-value in between the critical values. If it is true, solution is an “and” If it is not true, solution is an “or”
- 11. Example: Solve x2 + 2x 8
- 12. Your Turn! Solve 2x2 – x > 3