7.5 Systems Of Inequalities
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This document provides instructions for graphing linear inequalities on a coordinate plane. It explains that linear inequalities are graphed as lines but must be shaded according to the inequality sign. To graph an inequality, one must first put it in slope-intercept form, then graph the line as dashed or solid based on the inequality sign, and lastly shade the appropriate region indicated by the inequality sign.
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System of linear inequalities
System of linear inequalitiesmstf mstf
?
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 ¡Ü 100 (they have 100 bottles total)
3x + 5y ¡Ý 400 (they need at least $400)
Graph the regions defined by these inequalities and the overlapping region is the solution.Linear equations
Linear equationshireshgupta1997
?
This document discusses linear equations in one variable. It defines linear equations as those involving single variables with the highest power being 1. It presents rules for solving linear equations, including adding, subtracting, multiplying, or dividing the same quantity to both sides. Transposition as a method is explained, where terms change signs when shifted between sides of an equation. Examples of solving linear equations are provided. The document also discusses applying linear equations to word problems by setting up the equation based on the problem and solving for the unknown variable. Several examples of solving word problems involving linear equations are presented.Linear Inequalities
Linear InequalitiesBitsy Griffin
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This document discusses how to graph linear inequalities on a coordinate plane. It explains that less than and greater than signs are represented by dashed lines, while less than or equal to and greater than or equal to signs use solid lines. It notes that inequalities must be written in slope-intercept form before graphing and that the shading depends on whether the inequality involves less than or greater than. Specifically, y greater than inequalities are shaded above the line and y less than inequalities are shaded below the line. Practice graphing many problems to fully understand how to determine the shading direction.Linear Equations
Linear Equationsrfant
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Students learn to define and identify linear equations. They also learn the definition of Standard Form of a linear equation.
Students also learn to graph linear equations using x and y intercepts.Linear Equation with One Variable (LEOV)
Linear Equation with One Variable (LEOV)SMPK Penabur Gading Serpong
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3) The final section directs the reader to solve two sample linear equations with one variable, tying together the concepts discussed in the document. Linear Equation In one variable class 7
Linear Equation In one variable class 7Poulami Choudhury
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This document defines and provides examples of linear equations in one variable. It explains that a linear equation is an equation that can be written in the form ax + b = c or ax = b, where a, b, c are constants and a ¡Ù 0. Examples of linear equations given include 3x + 9 = 0 and 7x + 5 = 2x - 9. The document also discusses how to determine if a value is a solution to a linear equation by substitution and simplification. Steps for solving linear equations are provided, which include isolating the variable using inverse operations like addition/subtraction and multiplication/division.linear equation
linear equationkanishkaanandkv
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1) The document discusses linear equations in two variables, including defining their form as ax + by = c, explaining that they have infinitely many solutions, and noting that their graphs are straight lines.
2) Specific topics covered include finding solutions, drawing graphs, identifying equations for lines parallel to the x-axis and y-axis, and providing examples of writing and solving linear equations.
3) The summary restates the key points about the properties of linear equations in two variables, such as their graphical and algebraic representations.Chapter 3. linear equation and linear equalities in one variables
Chapter 3. linear equation and linear equalities in one variablesmonomath
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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 ¡Ü 40
3) Isolate x by undoing the operations
4) Write the solution set
The solution is 0 ¡Ü x ¡Ü 7
9.3 Foil Method
9.3 Foil Methodramjdram
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The document provides instructions for students to review multiplying binomials using the distributive property and FOIL method. It asks the student to recall what a monomial and binomial are, and explains that the distributive property and exponent rules can be used simultaneously. Examples are provided from pages 430, 438-441, and 436 for the student to practice multiplying binomials using the FOIL method, which stands for First, Outer, Inner, Last terms when multiplying two binomial expressions.9.1 Adding And Subtracting Polynomials
9.1 Adding And Subtracting Polynomialsramjdram
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This document provides instructions for adding and subtracting polynomials. It defines a monomial as a single term and a polynomial as multiple monomials separated by plus or minus signs. The degree of a polynomial is the highest exponent among its terms. When writing polynomials in standard form, the term with the highest exponent goes first, followed by terms in descending exponent order. Some polynomials have special names. To add polynomials, add the coefficients of like terms, ignoring the exponents. Subtracting polynomials follows the keep, change, change rule and watches the subtraction signs. Practice problems are provided.8.4 Exponents
8.4 Exponentsramjdram
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The document provides a summary of rules for negative and zero exponents:
1) Negative exponents: flip the base and exponent, changing the exponent to positive.
2) Zero exponents: any variable, monomial, or constant raised to the power of 0 is always equal to 1.
3) You must know the order of operations when rules are combined, such as changing a negative exponent first before other rules.8.3 Exponents
8.3 Exponentsramjdram
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The document provides an overview of lessons on laws of exponents for dividing monomials. It discusses the rule for quotients of powers which means subtracting exponents when dividing like bases. It also notes to treat coefficients like numbers and divide them. Examples are provided for using variables and fractions as bases when dividing monomials. Students are instructed to practice problems applying all exponent rules.8.2 Exponenrts
8.2 Exponenrtsramjdram
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This document provides a summary of key concepts from an algebra warm up on exponents including: when multiplying like bases you add exponents, when multiplying monomials you multiply coefficients and add exponents of like bases, when raising a power to a power you multiply the exponents, an exponent applied outside parentheses applies to everything inside, and that when raising a negative number to an exponent, an even exponent gives a positive answer while an odd exponent gives a negative answer. It directs the reader to specific pages to review these exponent rules and examples.8.1 Exponent Multiplication
8.1 Exponent Multiplicationramjdram
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This document provides instructions on multiplying monomials using the laws of exponents. It defines key terms like base and exponent. It explains that when multiplying like bases, you add the exponents. It provides the steps to multiply monomials as either: 1) multiplying the coefficients and adding the exponents of like bases, or 2) dropping parentheses, rearranging coefficients and variables, then multiplying like bases by adding exponents and simplifying. Examples are provided for students to practice this skill.7.6 Classic Puzzles
7.6 Classic Puzzlesramjdram
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This document discusses solving word problems involving two unknown variables through systems of equations. It provides examples of different types of puzzles including age, travel rate with or against current/wind, coin values, chemical solutions, and number digits. It emphasizes defining the two variables, writing two related equations, and using methods like elimination to solve systems of equations to find the unknown values. Students are instructed to practice setting up the equations for several sample problems without solving them.Ch. 7 Review
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7.4 Classifying Systemsramjdram
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This document discusses different types of systems of equations and their solutions. It introduces inconsistent systems which have no solution because the lines are parallel. Consistent and independent systems have a single solution, which is what previous examples focused on. Consistent and dependent systems can have either an infinite number of solutions if the lines are the same, or no solution if the lines are parallel. Examples of each type of system are provided.7.3 Elimination Method
7.3 Elimination Methodramjdram
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The document discusses the elimination method, which is the third method for solving systems of equations. It involves adding the equations together to cancel out one of the variables, allowing it to be solved for. The key steps are to line up the equations so that one variable has the same coefficient but opposite sign, then add the equations and solve for that variable before substituting back into one equation to solve for the other variable. The reader is directed to an activity on page 332 to practice applying these steps of the elimination method.7.1 Graphing Systems Of Equations
7.1 Graphing Systems Of Equationsramjdram
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This document provides instructions for graphing systems of equations. It explains that a system of equations involves two equations with two variables that share a common solution point where the lines cross on the graph. It notes that to graph a system, one should graph each line on the same axes and label the point where they intersect as the solution to the system. It also provides a brief review of how to graph individual lines by first rewriting the equation in slope-intercept form and then determining points by either locating the y-intercept or using a t-chart to substitute values of x and solve for y.7.2 Substitution Method
7.2 Substitution Methodramjdram
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The document discusses the substitution method for solving systems of equations, which allows you to find the exact solution without graphing. It explains that the substitution method involves solving one equation for one variable and substituting that value into the other equation to solve for the remaining variable. An example problem demonstrates substituting the value of x from one equation into the other equation to solve for y.6.5 Absolute Value Inqualities
6.5 Absolute Value Inqualitiesramjdram
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Absolute value equations have two answers because the absolute value bars mean the quantity inside can be positive or negative. To solve absolute value equations, set up two cases - one where the quantity is positive and one where it is negative - and solve each case separately to find the two answers. Absolute error refers to the absolute value of the difference between an actual measurement and a specified value. Absolute value inequalities allow you to find the greatest and least acceptable values for a variable.6.4 Absolute Value
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This document provides instructions for students to complete algebra warm up problems involving absolute value functions. Students are directed to complete problems 75 to 83 on page 299 using laminated graph paper, and problems 17 to 35 on page 298 after reviewing the properties of absolute value, including that it represents distance from 0 and acts like parentheses.6.3 Compound Inequalities
6.3 Compound Inequalitiesramjdram
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The document provides instructions for students to complete homework questions for 6.1 to 6.2 by bringing their paper to the front of the class for 20 minutes without talking or asking questions. It then instructs students to read pages 289 to 293 about compound inequalities, which combine two inequalities using "AND" or "OR". It explains that an "AND" inequality involves two situations happening at the same time and being met simultaneously, while an "OR" inequality allows either situation to occur.6.2 Multistep Equations
6.2 Multistep Equationsramjdram
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The document provides instructions for students to complete algebra problems from specific pages in their textbook. It asks students to complete problems 72 through 90 on page 288, as well as problems 1 through 14 on page 753. It also reminds students about solving inequalities and using the distributive property, inverse operations, and combining like terms when solving multi-step equations.6.1 Solving Inequalities
6.1 Solving Inequalitiesramjdram
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Solving inequalities is similar to solving equations, except the inequality sign changes when multiplying or dividing by a negative number. When graphing inequalities on a number line, locate the answer on the number line and use open or closed circles depending on whether the inequality uses < or >, and shade in the appropriate side.5.5 Point Slope and Standard Form
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This document provides instructions for students to complete algebra warm up problems involving graphing lines from equations. It outlines three methods for graphing lines: using the slope and y-intercept, using a T-chart, and manipulating equations into slope-intercept form. Students are informed they will be tested on using both the slope-intercept method and the T-chart method for graphing lines from equations. They are also told they will need to learn how to manipulate equations in standard form and point-slope form into slope-intercept form.5.4 Slope Intercept Form
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The document provides an overview of slope-intercept form for linear equations. It defines slope as the constant rate of change and the y-intercept as the y-value when x=0. Slope-intercept form is given as y=mx+b, where m is the slope and b is the y-intercept. This form can be used to graph lines, find points, and describe situations with a constant rate of change. Understanding slope-intercept form is important for showing what y-values occur at different x-values for a linear relationship.A Framework for Model-Driven Digital Twin Engineering
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This document provides instructions on multiplying monomials using the laws of exponents. It defines key terms like base and exponent. It explains that when multiplying like bases, you add the exponents. It provides the steps to multiply monomials as either: 1) multiplying the coefficients and adding the exponents of like bases, or 2) dropping parentheses, rearranging coefficients and variables, then multiplying like bases by adding exponents and simplifying. Examples are provided for students to practice this skill.7.6 Classic Puzzles
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This document discusses solving word problems involving two unknown variables through systems of equations. It provides examples of different types of puzzles including age, travel rate with or against current/wind, coin values, chemical solutions, and number digits. It emphasizes defining the two variables, writing two related equations, and using methods like elimination to solve systems of equations to find the unknown values. Students are instructed to practice setting up the equations for several sample problems without solving them.Ch. 7 Review
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