# Ready To Go On? Skills Intervention 2-1 Solving Linear Equations and Inequalities

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1 A Read To Go n? Skills Intervention -1 Solving Linear Equations and Inequalities Find these vocabular words in Lesson -1 and the Multilingual Glossar. Vocabular equation solution of an equation linear equation in one variable identit contradiction inequalit Solving Equations with Variables on Both Sides Solve To get the constant on one side of the equation, subtract 10 from both sides of the equation. To get the variable on one side of the equation, add to both sides of the equation. To isolate, divide both sides of the equation b. Solve for. Solving Inequalities Solve and graph. 3 ( 8) 15 3 ( 8) Distribute 3 to both terms in the parentheses. 15 Multipl. 1 1 Subtract 1 from both sides to isolate the variable. Graph the solution. A(n) Divide both sides b 3 to isolate. Do ou need to reverse the 3 3 inequalit smbol? Solve for. circle should be used and the arrow should point to the Test 0 in the original inequalit. Does our solution check? 3 ((0) 8) 15 3 (8) Holt Algebra

2 A Read to Go n? Problem Solving Intervention -1 Solving Linear Equations and Inequalities Solving a linear equation requires isolating the variable on one side of the equation b using the properties of equalit. Isabella is paid a salar of \$00 per month plus a commission of % of the sales price of each house she sells. Find the value of the houses Isabella must sell in one month to earn \$00. Understand the Problem 1. What are ou tring to determine?. What two things make up Isabella s monthl income? 3. What part of Isabella s monthl income is alwas the same, or constant?. What part of Isabella s monthl income changes each month? Make a Plan 5. What percentage of each house sale does Isabella earn?. What is the decimal equivalent of %? % 7. If Isabella sells no houses in one month, how much does she earn? 8. If Isabella sold a house for \$100,000, how much commission would she earn? % of \$100, (\$100,000) \$ 9. If Isabella sold onl one house for \$100,000 during the month, write a numerical epression to show much mone she would earn for the month. ( ) 10. How much mone is Isabella hoping to earn? Solve 11. If h is the value of the houses Isabella sells, represent the situation with an equation. 00 ( )h 1. Solve the equation for h. 00 h h h 13. Isabella must sell \$ worth of houses in one month to earn \$00. Look Back 1. Substitute our value for h into the original equation from Eercise ( ) 00. Does the left side equal the right side? Holt Algebra

3 A Read to Go n? Skills Intervention - Proportional Reasoning Find these vocabular words in Lesson - and the Multilingual Glossar. Vocabular ratio proportion rate similar indirect measurement Solving Proportions Solve each proportion. A. 8 9 When a proportion contains a variable, use cross products to solve for the ( ) Set the cross products equal. Mutlipl. Divide b to solve for. B. 7 Solve for ( ) ( ) Set the cross products equal. 3 Mutlipl. 3 Divide b to solve for. Simplif the fraction. C (3) 1.8( ) Set the cross products equal Solve for. Multipl. The product of a negative and a positive number is. Divide b to solve for. 5 Holt Algebra

4 A Read to Go n? Problem Solving Intervention - Proportional Reasoning To measure an object that cannot be easil measured, use indirect measurement. A cell tower casts a 30-ft shadow at the same time a 10-foot street sign casts a -ft shadow. How tall is the cell phone tower? Understand the Problem 1. Label the diagram with the given information.. How long is the cell tower s shadow? 3. How tall is the street sign?. What does h in the diagram represent? h ft ft ft Make a Plan 5. Since the triangles formed b the shadows are similar, use a to find h, the height of the cell phone tower.. The height of the cell phone tower corresponds to which part of the street sign? 7. The length of the cell phone shadow corresponds to which part of the street sign? 8. Complete the proportion: Height of cell phone tower Length of shadow of cell phone tower Length of shadow of street sign Solve h 9. Solve the proportion for h. 10. How tall is the cell phone tower? h h ( )( ) Set Cross Products equal. h h Multipl. Look Back 11. Substitute the value for h from Eercise 9 into the proportion h If the cross products are equal, the value for h is correct. ( ) 300. Does our answer check? Holt Algebra

5 A Read to Go n? Skills Intervention -3 Graphing Linear Functions Find these vocabular words in Lesson -3 and the Multilingual Glossar. Vocabular linear function slope -intercept -intercept slope-intercept form Graphing Lines Using the Intercepts Find the intercepts of 1 and graph the line Find the -intercept. Find the -intercept. Substitute 0 for. ( ) 1 Substitute 0 for. ( ) 1 Multipl. 1 Multipl. 1 Divide. 1 Divide. 1 Solve for. Solve for. The -intercept is the point (, 0). The -intercept is the point (0, ). Plot the two points ou found on the graph. Draw a straight line through the points. Graph Functions in Slope-Intercept Form Write the function in slope-intercept form. Then graph the function. First, solve for. Subtract from both sides. Divide b to isolate. 8 3 Simplif. What is the coefficient of? What is the constant? when 0: (0, Is the slope positive or negative? This is the slope of the line. This is the -intercept, or the -coordinate ). Plot this point on the graph. So, starting at the point of the -intercept, move unit(s) and to the right one unit. Draw a straight line through the two points. 7 Holt Algebra

6 A Find this vocabular word in Lesson - and the Multilingual Glossar. Writing Equations of Lines Write the equation of the line through (1, ) and (, 8) in slope-intercept form. Let ( 1, 1 ) be (1, ) and (, ) be (, ). Complete to find the slope of the line. m 1 1 Although ou can choose either point, substitute for in the equation of a line, m b, the -coordinate of (, 8) and for, the -coordinate of (, 8). m b m ( ) b Substitute values for and. ( ) b Substitute the value of m, the slope of the line. b Multipl. Subtract from both sides to solve for b. b Solve for b. Rewrite m b using m and b. Read to Go n? Skills Intervention - Writing Linear Functions Vocabular point-slope form Writing Equations of Parallel and Perpendicular Lines Write the equation of the line through (3, 7) and parallel to 1 in slope-intercept form. 3 What do ou know about the slopes of parallel lines? So, the slope of the line parallel to 1 is equal to. 3 Substitute for in the equation of a line, m b, the -coordinate of (3, 7) and for, the -coordinate of (3, 7). m b m( ) b Substitute values for and. ( ) b Substitute the value of m, the slope of the line. b Multipl. Add to both sides to solve for b. b Solve for b. Rewrite m b using m and b. 8 Holt Algebra

7 A Read to Go n? Skills Intervention -5 Linear Inequalities in Two Variables Find these vocabular words in Lesson -5 and the Multilingual Glossar. Vocabular linear inequalit boundar line Graphing Linear Inequalities Solve for in. Then graph. Add to both sides to isolate the variable. Solve for. What is the boundar line? Is the boundar line part of the solution? Should the boundar line be solid or dashed? Draw the boundar line on the graph. Should the region above or below the boundar line be shaded? Choose a value for, such as 0. Substitute this value into the inequalit. Substitute 0 for. Does the point satisf the inequalit? Graphing Linear Inequalities Using Intercepts Solve for in 1. Then graph. Find the -intercept b substituting 0 for. Find the -intercept b substituting 0 for. 1 1 (0) 1 (0) , so the -intercept is (, 0). 1, so the -intercept is (0, ). Use the - and -intercepts to draw the boundar line. Should the boundar line be solid or dashed? Substitute (0, 0) in the inequalit for and. 1 (0) (0) 1 If this point makes the statement true, shade the region containing the point. If not, shade the opposite region. 8 9 Holt Algebra

8 A When graphing a real-world application of an inequalit graph onl the part of the plane that includes realistic solutions. Adam s school is holding its annual musical. Tickets to evening shows cost \$.50 and tickets to afternoon shows cost \$.00. The school needs to make at least \$0 to cover epenses. Write and graph an inequalit for the number of each tpe of ticket that must be sold to make a profit. Understand the Problem 1. What are the two prices of the tickets?. How much mone does the school need to make to cover epenses? Make a Plan Read to Go n? Problem Solving Intervention -5 Linear Inequalities in Two Variables 3. If is the number of evening tickets, what do ou need to multipl b to find the amount the school makes b selling evening tickets?. If is the number of afternoon tickets, what do ou need to multipl b to find the amount the school makes b selling afternoon tickets? Solve 5. Complete the inequalit to describe the situation Find the intercepts of the boundar line. -intercept: -intercept:.5(0) 0.5 (0) The -intercept is (0, ). The -intercept is (, 0). 7. Plot the intercepts and draw a line through the two points. 8. Should ou shade above or below this boundar line? Look Back 9. Test a point, such as (0, 0) in the inequalit from Eercise ( ) ( ) 0 Substitute 0 for and 0 for Is the inequalit true? Is the graph shaded correctl? 30 Holt Algebra

9 A Read To Go n? Quiz -1 Solving Linear Equations and Inequalities Solve ( 8) (3 ) ( ) Solve and graph ( 3) ( 5) 8. 7 (5 3) Joe has saved \$ to bu a mountain bike that costs \$8. Joe gets paid \$0 for each lawn he mows. How man lawns must Joe mow to have enough mone to bu the bike? Proportional Reasoning Solve each proportion A tree casts a -foot shadow at the same time that a 1-foot pole casts a -ft shadow. How tall is the tree? 31 Holt Algebra

10 A Read to Go n? Quiz continued -3 Graphing Linear Functions Find the intercepts and graph each line Write each function in slope-intercept form. Then graph the function Writing Linear Functions Write an equation in slope-intercept form for each line. 19. through (3, 11) and (5, 19) 0. slope 1 and through (3, 5) 3 1. parallel to 5 3 and through (, 1). perpendicular to 3 9 and through (, 1) -5 Linear Inequalities in Two Variables Solve for in each inequalit. Then graph Holt Algebra

11 B Read to Go n? Enrichment Equations of Lines Determine the letter of the equation of a line from the table below that represents the same line as the given equation. A E...5(.3) I..5 B. (3) 5 9 ( (7)) F C G D. 0.5 (.3 ) H. (1) ( (1)) J ( ) ( 5) 9. (1) ( 0) 5 ( ) Holt Algebra

12 B Read to Go n? Skills Intervention - Transforming Linear Functions Translating and Reflecting Linear Functions Let g ( ) be the indicated transformation of f ( ). Write the rule for g ( ). f ( ) 1; vertical translation 3 units up Does a vertical translation change the input values or the output values? What number is being added to each value? g( ) f () Replace f() with the function given. g( ) ( ) Simplif the final function. g( ) Stretching and Compressing Linear Functions Let g ( ) be the indicated transformation of f ( ). Write the rule for g ( ). f ( ) 5; vertical compression b a factor of 1 How does a vertical compression change the graph of a function? Does a vertical compression change the input values or the output values? Multipl f ( ) b the factor of the compression. g () 5 Simplif the function. g () Combining Transformations of Linear Functions Let g ( ) be the indicated transformation(s) of f ( ). Write the rule for g ( ). f ( ) 8; horizontal stretch b a factor of followed b a horizontal translation to the right units What is the first transformation? Do the input values or the output values change? What is the function after the first transformation? h( ) f 1 b What is the second transformation? How do ou translate a function horizontall to the right? h ( ) 8 Perform the second transformation to find g(). g ( ) h ( ) g ( ) g ( ) 3 Holt Algebra

13 B Read to Go n? Skills Intervention -7 Curve Fitting with Linear Models Find these vocabular words in Lesson -7 and the Multilingual Glossar. Vocabular regression correlation line of best fit correlation coefficient Finding the Slope of a Line Find the slope of each line. Then write the equation that fits the data. A. Does the line slant upward or downward? Predict if the slope is positive or negative. Select one point on the line and call it ( 1, 1 ). (10, ) Select another point on the line and call it (, ). (50, ) Substitute these ordered pairs into the slope formula and solve for m. m m ( 1 ) Use the point-slope form. ( ) Substitute the values for 1, 1, and m. Distribute. Add to isolate. Simplif. B. Does the line slant upward or downward? (0, 8) 50 (1, 0) Predict if the slope is positive or negative. Select one point on the line and call it ( 1, 1 ). (0, ) Select another point on the line and call it (, ). (1, ) Substitute these ordered pairs into the slope formula and solve for m. m m ( 1 ) Use the point-slope form. ( ) Substitute the values for 1, 1, and m. Distribute. Add to isolate. Simplif. 35 Holt Algebra

14 B Read to Go n? Problem Solving Intervention -7 Curve Fitting with Linear Models A scatter plot is helpful in understanding the relationships between two variables. A particular compan has offices in the United States and in Ital. Job applicants must be able to read and speak both English and Italian. As part of the application process, prospective emploees must take a test on their knowledge of Italian. The personnel office compared the number of ears applicants studied Italian to their test scores. Make a scatter plot of the data, and then sketch a line of best fit and find its equation. Years of Stud Test Scores Understand the Problem 1. What two variables does the data describe?. What three things are ou asked to do? Make a Plan 3. Which variable should be plotted as the independent variable (input)?. Which variable should be plotted as the dependent variable (output)? Solve 5. How man data points can ou plot from the data? Plot these points on the grid provided.. Is the correlation positive (upward) or negative (downward)? 7. Draw a line that splits the data evenl above and below the line. What are two points on the line? (, ); (, ) 8. Use two points on the line, such as (, 50) and (5, 88) to find the slope of the line. m Use the point (, 50) and the slope from Eercise 8 to write the equation of the line in point slope form. 1 m( 1 ) ( ) Look Back 10. Tr related points in the equation from Eercise 9 to see if the answer is reasonable. For eample, substitute 3 for. Is the output value near the other points on the scatter plot? 3 Holt Algebra Test Scores Years of Stud

15 B Read to Go n? Skills Intervention -8 Solving Absolute-Value Equations and Inequalities Find these vocabular words in Lesson -8 and the Multilingual Glossar. Vocabular disjunction conjunction absolute value Solving Absolute-Value Equations Solve each equation. A. 1 1 or 1 Rewrite the absolute value as a disjunction What are the possible values of? B Divide both sides of each equation b. or Add to both sides of the equation. Divide each side of the equation b. So, or. Solving Absolute-Value Equations with Disjunctions Solve the inequalit 3 1. Then graph the solution. 3 or 3 Rewrite the absolute value as a disjunction. Subtract from both sides of each equation Divide both sides of each equation b 3. { or } Graph the solution. Should the circles be empt or solid? If is less than a number, draw an arrow to the left of the number. If is greater than a number, draw an arrow to the right of the number. 37 Holt Algebra

16 B Find this vocabular word in Lesson -9 and the Multilingual Glossar. Translating Absolute-Value Equations Translate f ( ) so that the verte is at the given point. Then graph. A. (0, 8) Let (0, 8) be (h, k). In the absolute-value function below, substitute h and k with the given point. g ( ) h k g ( ) Substitute values for h and k. g ( ) Simplif. Recall that the general forms for translations are: Vertical: g () f ( ) k Horizontal: g ( ) f ( h) Does the new graph have a horizontal shift from f ( )? If so, b how man units and in which direction? Does the new graph have a vertical shift from f ( )? If so, b how man units and in which direction? Shift and draw the graph accordingl. Is the verte of the new graph at (0, 8)? B. (1, 5) Read to Go n? Skills Intervention -9 Absolute-Value Functions Let (1, 5) be (h, k). In the function below, substitute h and k with the given point. g( ) h k Does the new graph have a horizontal shift from f ( )? If so, b how man units and in which direction? Does the new graph have a vertical shift from f ( )? If so, b how man units and in which direction? 10 8 Vocabular absolute-value function Shift and draw the graph accordingl. Is the verte of the new graph at (1, 5)? Holt Algebra

17 B Read to Go n? Problem Solving Intervention -9 Absolute-Value Functions To echange dollars for francs at the bank, the bank charges a commission in dollars equal to the echange rate times the difference of dollars and francs. For ever dollar echanged, the customer will receive 1.0 francs. For ever franc echanged, the customer will receive 0.80 dollars. So if a customer echanged \$100 for 10 francs, the difference of dollars and francs is 0. a. What function represents the commission the bank earns for echanging dollars and francs? b. Graph the function. Understand the Problem 1. Upon what two variables does the commission depend?. Can the difference of dollars and francs be negative? Wh? 3. Can the commission be negative? Wh? Make a Plan. How can ou write the function so that the difference of dollars and francs alwas results in a positive commission? 5. If is the difference of dollars and francs, and r is the echange rate, what operation do ou use to determine the commission? Solve. Write an absolute value function to describe the 0 18 commission. 7. Graph the function on the grid given that the echange rate, r is 0% or Look Back 8. Check the graph. Is the commission alwas positive? 9. If ou echange \$100 for 10 francs and the echange rate is 0% what commission does the bank earn?. Is this a reasonable amount of mone? 39 Holt Algebra

18 B Read to Go n? Quiz - Transforming Linear Functions Let g () be the indicated transformation(s) of f (). Write the rule for g (). 1. f () 3; vertical translation 3 units down. f () ; vertical stretch b a factor of 3. f () ; horizontal compression b a factor of 1 followed b a horizontal translation left 8 units. f () ; horizontal translation units right followed b a vertical compression b a factor of Curve Fitting with Linear Models 5. A student has kept track of the relative humidit and the apparent room temperature. The results are shown in the table below. Relative Humidit (%) Apparent Room Temperature, (F) Apparent Temperature ( F) Relative Humidit (%) a. Draw a scatter plot of the data using relative humidit as the independent variable. b. Use our graphing calculator to find the correlation coefficient and the equation of the line of best fit for the data. What does the slope of the best fit mean for this data? c. Use our equation to predict the apparent room temperature at a relative humidit of 5%. 0 Holt Algebra

19 B Read to Go n? Quiz continued -8 Solving Absolute-Value Equations and Inequalities Solve each equation Solve each inequalit. Then graph the solution Absolute-Value Functions Translate f () so that the verte is at the given point. Then graph. 1. (0, 3) 15. (, ) 1. (5, 0) A cereal compan fills ever bo with 8 ounces of cereal. The compan allows each bo of cereal to be within a tolerance of 0.5 ounces. What is an epression for the actual weight of the boes? 1 Holt Algebra

20 B Read to Go n? Enrichment Scatter Plots Match the correlation coefficient to the data it most likel describes. A B C D 1. r 0.9. r 0 3. r r 0.97 Arrange the correlation coefficients in order from the weakest correlation to the strongest , 0.9, 0.15, , 0., 0.98, , 0.010, 0.011, , 0.909, 0.099, Identif each statement as true or false. 9. A scatter plot in which there is no relation between the data has a correlation coefficient close to Some scatter plots have a correlation coefficient that is greater than 1, which indicates an even stronger relation between the data values. 11. A correlation coefficient close to 1 indicates a relation with a strong linear trend with a negative slope. Holt Algebra

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