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
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.
Linear equations
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.
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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.
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Students learn to define and identify linear equations. They also learn the definition of Standard Form of a linear equation.
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Linear Equation with One Variable (LEOV)
1) The document discusses linear equations with one variable (LEOV). It defines key terms like statements, open and closed sentences, equations, and the components of a linear equation with one variable.
2) Examples are provided to illustrate open sentences that can be made into closed sentences or equations by replacing variables with values. Exercises ask the reader to write open sentences as equations and solve simple equations.
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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 ≤ 40
3) Isolate x by undoing the operations
4) Write the solution set
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Recommended
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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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.
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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.
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Students also learn to graph linear equations using x and y intercepts.
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linear equation
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
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
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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
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
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.
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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.
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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.
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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
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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.
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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.
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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
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.
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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
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.
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