Solving inequalities
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This document discusses how to solve inequalities by treating them as equalities and then determining the set of numbers that make the inequality true. It explains that the solution to an inequality is a set or interval expressed visually by graphing. Examples are provided to demonstrate solving different types of inequalities algebraically and checking solutions by plugging numbers into the original inequality. Special rules are outlined for when multiplying or dividing across an inequality by a negative number.
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4.4 notes
This document discusses solving compound inequalities, which contain two inequalities joined by "and" or "or". It provides examples of writing compound inequalities from word phrases and graphing the solutions. Examples are given of solving compound inequalities containing "and" or "or" and graphing the solutions.
6.4 elimination using multiplication
This document provides instructions for solving systems of equations by elimination using multiplication. It explains that you should multiply one equation by a constant to create opposite terms, add the equations to eliminate one variable, solve the resulting equation for the remaining variable, and then substitute the solution back into one of the original equations to solve for the second variable. It then provides examples of solving six systems of equations using this elimination method and assigns additional practice problems as homework.
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This document discusses solving and graphing inequalities on a number line. It explains that solving inequalities is similar to solving other algebraic equations by manipulating the inequality signs in the same way as equal signs. When graphing the solutions, open circles should be used for "greater than" and "less than", while closed circles indicate "less than or equal to" and "greater than or equal to". An example problem walks through solving and graphing the inequality c + 7 < -11.
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This document discusses solving compound inequalities, which contain two inequalities joined by "and" or "or". It provides examples of writing compound inequalities from word phrases and graphing the solutions. Examples are given of solving compound inequalities containing "and" or "or" and graphing the solutions.
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This document provides instructions for solving systems of equations by elimination using multiplication. It explains that you should multiply one equation by a constant to create opposite terms, add the equations to eliminate one variable, solve the resulting equation for the remaining variable, and then substitute the solution back into one of the original equations to solve for the second variable. It then provides examples of solving six systems of equations using this elimination method and assigns additional practice problems as homework.
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This document discusses solving and graphing inequalities on a number line. It explains that solving inequalities is similar to solving other algebraic equations by manipulating the inequality signs in the same way as equal signs. When graphing the solutions, open circles should be used for "greater than" and "less than", while closed circles indicate "less than or equal to" and "greater than or equal to". An example problem walks through solving and graphing the inequality c + 7 < -11.
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The document provides instructions for an activity to measure angles of triangles and compare exterior angles to interior angles. Students are asked to measure and record the angles of three triangles, then compare each exterior angle to the remote interior angles using relation symbols. All comparisons showed the exterior angle is greater than the remote interior angle, allowing students to state the exterior angle inequality theorem.
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The document defines and provides examples of three measures of central tendency: median, mode, and mean. The median is the middle number in a data set arranged from lowest to highest. The mode is the number that occurs most frequently. The mean is the average, calculated by adding all values and dividing by the total number of values. Examples are given to demonstrate calculating each measure for different data sets.
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The document discusses subtracting integers. It explains that when subtracting integers with the same sign, the answer is always negative. When subtracting integers with different signs, subtract the smaller number from the larger number and the sign of the answer is the sign of the larger number. It provides examples of subtracting integers with both the same and different signs.
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The document defines integers and their properties like positive and negative numbers. It introduces rules for adding and subtracting integers, such as adding numbers with the same sign and subtracting numbers with different signs. It also explains how to use a number line to visualize adding integers and proves that subtracting a negative number is the same as adding a positive number.
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1) Quotient of powers - subtracting exponents when dividing terms with the same base.
2) Power of a quotient - finding the power of a term when the base is a fraction, by finding the powers of the numerator and denominator.
3) Zero exponent property - any number to the zero power is equal to 1.
4) Negative exponent property - a negative exponent is the same as the reciprocal of the positive exponent.
5) Several examples of simplifying expressions using these properties are provided.
7.1 multiplication property of exponents
1) This document discusses properties of exponents and monomials. It defines monomials as numbers, variables, or products of numbers and variables with no addition, subtraction, or variables in the denominator.
2) It covers the product of powers property, which states that when multiplying the same bases, you add the exponents. It also discusses raising a power to another power by multiplying the exponents.
3) The power of a product property is explained as finding the power of each factor and multiplying those exponents when the factors are multiplied. Examples of simplifying expressions using these properties are provided.
5.6 graphing inequalities
This document discusses graphing inequalities and provides examples of graphing different types of inequalities including linear, compound, and fractional inequalities. It also lists homework problems from page 320 problems 12 through 23 that involve graphing various inequalities on a coordinate plane.
6.3 solve by elimination
This document provides steps for solving systems of equations by elimination. It outlines the process of writing the system so like terms are aligned, adding or subtracting entire equations to eliminate a variable, and then substituting the result into one equation to solve for the remaining variable. It also provides examples of systems of equations to solve using this elimination method and assigns homework problems from the textbook.
6.2 substitution
This document contains 8 equations involving the variables x and y. The equations are: y = 2x + 1, 3x + y = -9, x + 2y = 6, 3x - 4y = 28, y = 2x - 4, -6x + 3y = -12, 2x - y = 8, and y = 2x - 3. The document also notes that homework problems 8 through 22 can be found on page 347.
6.1 solve systems of equations by graphing
This document discusses solving systems of equations by graphing. It defines a system of equations as a group of two or more equations. The solution to a system is the point where the graphs of the equations overlap or intersect. There are three possible cases: exactly one solution where the lines cross at one point; infinite solutions where the lines lie directly on top of each other; and no solution where the lines are parallel and never intersect. Graphing is presented as an easy way to solve systems by finding the point of intersection between the equations' graphs. Examples of systems to solve by graphing are provided.
4.4 parallel & perpendicular lines
This document provides information about parallel and perpendicular lines. It defines parallel lines as lines that never cross and have the same slope. Perpendicular lines are defined as lines that cross at a right angle, have slopes that are opposite reciprocals, and have a product of slopes equal to -1. The document contains examples of writing equations of lines parallel and perpendicular to given lines and determining whether pairs of lines are parallel, perpendicular, or neither. It concludes with homework problems from the text.
4.3b standard form
1) Standard form is the formal way to write the equation of a line as Ax + By = C, where A, B, and C are integers and A must be positive.
2) The document provides examples of writing a line equation in standard form and finding the slope, y-intercept, and x-intercept of lines written in standard form.
3) Students are instructed to complete practice problems involving writing equations in standard form, finding slopes and intercepts, and graphing lines.
4.3 point slope form
The document discusses writing equations in point-slope form. It provides the point-slope form equation y - y1 = m(x - x1) and examples of writing equations in point-slope form given a point and slope, finding the slope and a point from an equation, writing an equation in slope-intercept form, and finding the equation of a line given two points in point-slope form. The homework assigned is to complete practice problems writing and identifying equations in point-slope form.
4.2 writing equations in slope intercept form
The document discusses writing linear equations in slope-intercept form. It reviews that the slope-intercept form is y=mx+b, where m is the slope and b is the y-intercept. It provides examples of writing equations given the slope and y-intercept, given a point and the slope, and finding the equation of a line passing through two given points by first calculating the slope and then using one of the points to find the y-intercept.
4.1 slope intercept form
This document provides instructions for graphing linear equations in slope-intercept form using tables of values, discusses the components of the slope-intercept form equation (y=mx+b), gives examples of writing equations in slope-intercept form with given slopes and y-intercepts, discusses constant and zero-slope functions, and provides an example equation to graph along with assigned homework problems.
Rate of change & slope
This document discusses rate of change and slope. It defines rate of change as the ratio of change in y to change in x. Slope is defined as the ratio of the rise to the run between two points and can be calculated as the change in y over the change in x. Several examples are provided of calculating the slope between two points by using this slope formula. The document also provides examples of using a known slope and one point to solve for the missing coordinate of a second point.
Solving compound inequalities
This document provides an overview of solving compound inequalities involving "AND" and "OR" statements. It explains that for "AND" inequalities, the solution includes only values that satisfy both individual inequalities. For "OR" inequalities, the solution includes values that satisfy at least one of the individual inequalities. Examples of splitting compound inequalities into two or more separate inequalities are also presented.
Solving multi step inequalities
This document provides an overview of solving multi-step inequalities with examples of solving inequalities with variables and graphing the solution sets. It includes solving inequalities with addition, subtraction, multiplication, and division terms as well as compound inequalities. The homework assignments ask students to solve 34 multi-step inequality problems.
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Solving inequalities
- 1. Solving Inequalities Sections 5.1 & 5.2 page 285-297
- 2. You already know > ≥ < ≤ Greater than Greater than or equal to Less than Less than or equal to
- 3. But what does that mean? Inequality is a comparison The solution is a set of numbers More than just one thing in the solution Express as a set or an interval Visual solution – graph the set
- 4. What numbers make the inequality true? How do we express all the numbers that will work? 𝑥 > 5 graphical interval set (5, ∞) {𝑥|𝑥 > 5} 5
- 5. Solve the Inequality Pretend like the inequality is an equal sign and solve. Graph the solution set. Check your answers. 𝑥 − 3 > 7 +3 +3 𝑥 > 10 10 Pick a number in the solution set. 𝑙𝑒𝑡 𝑥 = 11 Plug it in and see if it works 11−3 > 7 8 > 7
- 6. 3𝑥 − 1 ≤ 11 Example: Solve the inequality and graph the solutions set.
- 7. When you change the sign you must change the sign. Multiply or divide across the inequality with a negative you must flip the inequality. You are changing the sign of the number so you must change the sign of the inequality to keep it true. 7 < 9 7 −2 9(−2) −14 − 18> Direction of the Inequality Changed
- 8. Example: Solve the inequality and graph the solution set. − 𝑐 6 ≥ 7
- 9. Example: Solve the inequality and graph the solution set. −28 ≤ −6𝑥
- 10. Example: Solve the inequality and graph the solution set. − 3 4 𝑟 ≤ −6
- 11. Homework Page 288 #12-29 Page 289 #12-29