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combine like ter hen combine lik e ter Guided Practice In Exercises 9–16, solve each equation for the unknown variable. 9. 4x = 144 11. x 4 = –7 13. –3x + 4 = 19 10. –7x = –7 12. 3 2x = 9 14. 4 – 2x = 18 15. 3x – 2(x – 1) = 2x – 3(x – 4) 16. 3x – 4(x – 1) = 2(x + 9) – 5x Section 2.2 Section 2.2 Section 2.2 — Manipulat... |
4 9 of the x points. If his overall grade for the test was 65, find the value of x. 20. Latoya takes 3 science tests. She scores 24%, 43%, and x% in the tests. Write an expression for her average percentage over the 3 tests. Her average percentage is 52. Calculate the value of x. Solve the following equations. Show al... |
3(23(23(23(23(2x – 5) + 4( h as sucsucsucsucsuch as h as – 2) = 12. – 5) + 4(x – 2) = 12. – 2) = 12. – 5) + 4( h as What it means for you: You’ll use the least common multiple to remove fractional coefficients from equations. Key words: coefficient least common multiple Expressions often seem more complicated if they ... |
3 8181818181 Guided Practice In Exercises 1–2, find the least common multiple of the denominators: 1. 1 9 x + 13 – In Exercises 3–6, solve the equations for the unknown variable. 3. 1 2 x – 1 = x 5. 1 10. 1 10 x – 3 = 1 4 x 6 or More Fe Fe Fe Fe Frrrrractions actions actions o or Mor k Out the LCM fCM fCM fCM fCM for o... |
3 x + 6 (m – 2) – 1 5 m = – 1 5 The sum of the measures of the angles of a triangle is 180°. In Exercises 9–11, find the value of x. 9. x 10. 11. 1 2( – 30) x 1 x 4 1 5 x x + 10 + 2 1 x + 5 10 x 2 5 x 2 5 x 12. Qiaofang bought a new car for $27,000. If the value of the car depreciates as 27,000 – 7100 3 n, where n is ... |
csucsucsucsuch as h as – 2) = 12. – 5) + 4(x – 2) = 12. – 2) = 12. – 5) + 4( h as What it means for you: You’ll use the least common multiple to remove fractional coefficients from equations. Key words: coefficient least common multiple ficients in ficients in actional Coef actional Coef FFFFFrrrrractional Coef ficient... |
the least common multiple. Solution Solution Solution Solution Solution There are 3 different denominators, so you need the LCM. List the prime factors, and use them to work out the LCM. Check it out: Use parentheses to group each numerator before you multiply by the LCM. 5 = 5 6 = 2 × 3 10 = 2 × 5 So LCM = 2 × 3 × 5 ... |
: Students simplify 4.0: Students simplify 4.0: Students simplify essions befororororore solving eeeeexprxprxprxprxpressions bef e solving e solving essions bef essions bef e solving e solving essions bef tions tions linear equa linear equa tions and linear equations tions linear equa linear equa in one variaariaariaar... |
idea is to multiply both sides of the equation by a large enough power of 10 to convert all decimals to integer coefficients. For example, 0.35 means “35 hundredths,” so you can write it So multiplying by 100 gives you the integer 35. 35 100. Example Example Example Example Example 11111 Solve the equation 0.35x – 12 ... |
+ 5 10. 0.25(x + 8) + 0.10(8 – x) + 0.05x = 3.60 y Higher Numbersssss ou Need to Multiply by by by by by Higher Number y Higher Number y Higher Number ou Need to Multipl Sometimes YYYYYou Need to Multipl ou Need to Multipl Sometimes Sometimes y Higher Number ou Need to Multipl Sometimes Sometimes In Examples 1 and 2 y... |
4(2x – 1) = 0.25(3x + 8) 2. 0.5x – 1 = 0.7x + 0.2 3. 0.3v + 4.2v – 11 = 1.5v + 4 4. 1.8(y – 1) = 3.1y + 2.1 5. 0.25(3x – 2) + 0.10(4x + 1) + 0.75(x – 1) = 4.55 6. 0.125(3b – 8) – 0.25(b + 5) – 0.2 = 0.2(2b + 7) 7. 0.4(x + 7) – 0.15(2x – 5) = 0.7(3x – 1) – 0.75(4x – 3) 8. 0.21(x – 1) – 0.25(2x + 1) = 0.5(2x + 1) – 0.6(4... |
The cost per minute to make a call is $0.05. If Meimei talks for (x + 20) minutes and the call costs $4.85, what is the value of x? 21. A cell phone plan charges $25 per month plus $0.10 per minute. If your monthly bill is $39.80, write and solve an equation to find out the number of minutes on your bill. 22. A moving... |
,,,,, in one v in one v inequalities in one v in one v – 2) = 12. – 2) = 12. – 5) + 4( – 5) + 4( h as 3(23(23(23(23(2x – 5) + 4( h as h as sucsucsucsucsuch as – 2) = 12. – 5) + 4(x – 2) = 12. – 2) = 12. – 5) + 4( h as 5.0:5.0:5.0:5.0:5.0: Students solv Students solv Students solveeeee Students solv Students solv ultist... |
Linear Equa Linear Equa Linear Equations tions tions Linear Equa Linear Equa “Applications” are just “real-life” tasks. In this Topic, linear equations start to become really useful. asksasks eal-Life”e”e”e”e” TTTTTasks asksasks eal-Lif tions are “Re “Re “Re “Re “Real-Lif eal-Lif tions ar tions ar Equa Equa tions of t... |
AlAlAlwwwwwaaaaays Be Gi ys Be Gi en the ys Be Gi Sometimes you’ll have to work out for yourself what the variables are, and decide on suitable labels for them. Example Example Example Example Example 22222 Juanita’s age is 15 more than four times Vanessa’s age. The sum of their ages is 45. Set up and solve an equation... |
is the midpoint of the line segment shown. Find the value of x and the length of the entire line segment. 6x – 11 4x + 23 M 3. The perimeter of the rectangular plot shown below is 142 feet. Find the dimensions of the plot. (6 – 1) ft x (4 + 2) ft x 4. The sum of the interior angles of a triangle is 180°. Find the size... |
e Linear Equations About Mone tions e Linear Equa asks In Coin If you have a collection of coins, there are two quantities you can use to describe it — how many coins there are, and how much they are worth. In coin tasks you’ll have to use both quantities. Example Example Example Example Example 11111 Jazelle has a co... |
tickets multiplied by the cost of each ticket. So 4.50x = amount raised from adults (in dollars) 5x(2.50) = amount raised from children Write an equation showing that the sum of these amounts is $3009: 4.50x + 5x(2.50) = 3009 17x = 3009 x = 177 So 177 adult tickets and 5 × 177 = 885 children’s tickets were sold. Indep... |
each kind did he buy with $29.75? 7. Martha bought some baseball uniforms for $313 and had them shipped to her. If the baseball uniforms cost $23.75 each and the shipping was $4.25 for the whole order, how many baseball uniforms did Martha buy? 8. Fifteen children’s books cost $51.25. Some were priced at $2.25 each, a... |
ound Up ound Up RRRRRound Up ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up To answer these questions, you need to use (i) the number of items, and (ii) their value. Then, when you have set up your equation and solved it, be sure to give your final answer in the form asked for in the problem. 9494949494 ... |
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Solution Solution Solution Call the first (smallest) even integer x. Then you can write down an expression for the other even integers in terms of x. 1st even integer = x 2nd even integer = x + 2 3rd even integer = x + 4 But the sum of the three even integers is 48, so x + (x + 2) + (x + 4) = 48. Now you can rearrange... |
must be 21, 25, and 29. Once you’ve completed the problem, do a quick answer check — add the integers together and see if you get what you want: 1st integer = v = 17 2nd integer = v + 4 = 21 3rd integer = v + 8 = 25 4th integer = v + 12 = 29 92 9696969696 Section 2.5 Section 2.5 Section 2.5 — Consecutive Integer Tasks... |
Practice 1. Three consecutive integers have a sum of 90. Find the numbers. 2. Find the four consecutive integers whose sum is 318. 3. Find three consecutive integers such that the difference between three times the largest and two times the smallest integer is 30. 4. Find three consecutive integers such that the sum o... |
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ted TTTTTasks AgAgAgAgAge-Re-Re-Re-Re-Relaelaelaelaelated asksasks asksasks ted ted ted Age tasks relate ages of different people at various time periods — in the past, the present, or the future. asksasks ted TTTTTasks asksasks ted AgAgAgAgAge-Re-Re-Re-Re-Relaelaelaelaelated ted ted Your age at any point in your life... |
contin Example 1 contin Example 1 contin Jorge’s future age = x + 20 Charles’s future age = (x + 7) + 20 = x + 27 Now use the information from the question to combine these expressions into an equation. The sum of their ages in 20 years will be 81, so: (x + 20) + (x + 27) = 81 Now solve your equation for x: (x + 20) +... |
In 15 years’ time, the father will be twice as old as his daughter. What are their current ages? Solution Solution Solution Solution Solution Present You are not given the daughter’s age, so call it x. The father is three times as old — three times x. Daughter’s age = x Father’s age = 3x Future (in 15 years) The daugh... |
who are all different ages. There is a 2 year difference in age between the oldest and youngest. Juan is not as old as Jose, but he is older than Keisha. If the sum of their ages is 36, how old is the oldest child? 2. Sonita’s father is three times as old as she is now. In ten years, her father will be twice as old as... |
ound Up ound Up Age tasks are just another example of real-life equations. As always, you have to set up an equation from the information you’re given, then solve the equation. Your answer is only complete when you include the actual ages of the people involved. 102102102102102 Section 2.5 Section 2.5 Section 2.5 — Co... |
Speed, The quantities of distance, time, and speed are related by a formula. Speed is the distance traveled per unit of time — for instance, the distance traveled in one second, one hour, etc. speed = distance time The units of speed depend on the units used for the distance and time. For example, if the distance is i... |
Solv ou Kno rite Do Motion tasks normally involve two objects with different speeds. Example Example Example Example Example 11111 Tim drives along a road at 70 km/h. Josh leaves from the same point an hour later and follows exactly the same route. If Josh drives at 90 km/h, how long will it take for Josh to catch up ... |
)sruohni( deepS )hpmni( ecnatsiD )selimni( = speed × time lehcaR eniarroL x x+1 3 55 54 55 x x+⎛ ⎜⎜⎜ 45 ⎝ ⎞ ⎟⎟⎟⎟ ⎠ 1 3 When Rachel catches up, Rachel and Lorraine have traveled equal x+⎛ ⎜⎜⎜ distances, so 55x = 45 ⎝ 1 3 ⎞ ⎟⎟⎟⎟. So solve this equation to find x: ⎠ 55 x = 45 x + 45 3 10 = x x = 15 3 2 x = 1 1 2 So, Rach... |
things are traveling in opposite directions, you have to think carefully about the equation. Example Example Example Example Example 33333 Bus 1 leaves Bulawayo at 8 a.m. at a speed of 80 km/h. It is bound for Harare, 680 km away. Bus 2 leaves the Harare depot at 8 a.m., heading for Bulawayo along the same highway at ... |
Mrs. Ding are traveling. Independent Practice 1. A plane leaves Miami for Seattle at 9:00 a.m., traveling at 450 mph. At the same time another plane leaves Seattle for Miami, flying at 550 mph. At what time will the planes pass if the distance between Miami and Seattle is 3300 miles? 2. A jet leaves the airport travel... |
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interest rate of 10%. What is the return on her investment at the end of one year? Solution Solution Solution Solution Solution The formula you need is I = pr, where p = $5000. Convert the percent to a decimal: r = 10 100 = 0.10 Substitute in your values for p and r: I = pr = 5000 × 0.10 = 500 So Maria makes $500 in i... |
pay 10% annually and the savings account pays 8%, how much interest will he make over the year? Solution Solution Solution Solution Solution I = p1r1 + p2r2 where p1 = 6000, r1 = 0.1 (= 10%) p2 = 10,000 – 6000 = 4000, r2 = 0.08 (= 8%) I = p1r1 + p2r2 = (6000 × 0.10) + (4000 × 0.08) = 600 + 320 = 920 So Francis will ea... |
at 5%, and p2 the amount invested at 9%. Let p1 = x, then p2 = 12,000 – x. Also, r1 = 0.05 (= 5%), and r2 = 0.09 (= 9%). Substitute these values into the annual interest formula: I = p1 r1 + p2r2 = 0.05x + 0.09(12,000 – x) = 700 Now you can solve for x: 0.05x + 0.09(12,000 – x) = 700 5x + 9(12,000 – x) = 70,000 Get ri... |
how much money, to the nearest dollar, did he invest at 8.25% if he earned $1166.30 total interest in a year? est are Re Re Re Re Relaelaelaelaelatedtedtedtedted est ar est ar Inter Inter Amounts of Sometimes TTTTTwwwwwo o o o o Amounts of Amounts of Sometimes Sometimes Interest ar Amounts of Inter est ar Inter Amount... |
the one earning 15% interest. How much was invested in each account? 17. Lavasha invested $16,000 among two different accounts paying 10% and 12% in one year. If she earned twice as much interest in the account paying 12%, how much did she invest in each account? Independent Practice 1. In one year, an investment at 8... |
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of mixture × 100 The following example doesn’t involve a mixture, but it shows how the above formula can be used. Example Example Example Example Example 11111 If a 1 kg bag of granola consists of 15% raisins (by mass), how many grams of raisins are there in the bag? Solution Solution Solution Solution Solution The fo... |
20% fruit drink, concentration = 0.2 = y 15, where y is the volume of fruit juice. The amount of fruit juice in the 20% solution is y = 0.2 × 15 = 3 liters. Also, the total volume of the combined fruit drink is 5 + 15 = 20 liters. The total amount of fruit juice in the mixture is x + y = 0.5 + 3 = 3.5 liters. Therefor... |
a party punch. What percentage of the party punch is pineapple juice? 7. A 2 kg bag of walnuts is mixed with 3 kg of pecans, 4 kg of hazelnuts, and 1 kg of brazil nuts to make a 10 kg bag of mixed nuts. What percentage of the 10 kg bag of nuts is pecans and hazelnuts? 8. 473 ml of 70% rubbing alcohol is combined with ... |
ur e Mixtur MorMorMorMorMore Mixtur e Mixtur e Mixture e e e e TTTTTasks MorMorMorMorMore Mixtur asksasks asksasks e Mixtur e Mixtur e Mixtur Remember that a mixture is made up of only the original ingredients. Nothing can mysteriously appear in the final mixture that was not part of the original ingredients, and nothi... |
15x 0.5x = 100 x = 200 Therefore 200 gallons of the 65% solution are required. 116116116116116 Section 2.6 Section 2.6 Section 2.6 — Investment and Mixture Tasks Section 2.6 Section 2.6 Guided Practice 1. A pest control company has 200 gallons of 60% pure insecticide. To create a 70% pure insecticide solution, the comp... |
away x cm3 of the 30% solution and replaced it with 70% solution. Then 0.3(500 – x) + 0.7x = 0.4(500) 150 + 0.4x = 200 x = 125 So the pharmacist replaced 125 cm3 of the 30% acid solution. Section 2.6 Section 2.6 Section 2.6 — Investment and Mixture Tasks Section 2.6 Section 2.6 117117117117117 Guided Practice 4. A doc... |
( w Check it out: Notice that p1w1 + p2w2 = pw. Compare this to the formula on the previous page — it’s essentially the same. First write down what you know about the original ingredients and the final mixture (calling the quantity you need to find x, for example). Then relate a “before” total to the “after” total — s... |
kilogram? 11. A landscaper wants to make a blend of grass seed using 300 pounds of $0.40 per pound grass seed and another seed costing $0.75 per pound. How much of the $0.75 seed does the landscaper need to make a $0.60 per pound blend? ut the Same Maththththth ut the Same Ma ut the Same Ma xt — b xt — b ent Conte A D... |
per pound and grapes cost $1.10 per pound. Michael wishes to make a fruit tray using only apples and grapes that costs $1.00 per pound. If Michael has 8 lbs of apples, how many pounds of grapes are needed? 14. A roast coffee blend costing $6.60 per pound is made by mixing a bean that costs $2.00 per pound with another... |
ound Up ound Up ound Up Over the last couple of Topics you’ve seen lots of examples of mixture tasks. The most difficult part of a mixture task is setting up the equation — once you’ve done that it’s all a lot easier. 120120120120120 Section 2.6 Section 2.6 Section 2.6 — Investment and Mixture Tasks Section 2.6 Section... |
Speeds of asks ar ted This area of math is concerned with calculating how long certain jobs will take if the people doing the job are working at different rates. Example Example Example Example Example 11111 John takes 1 hour to deliver 100 newspapers, and David takes 90 minutes to deliver 100 newspapers. How long wou... |
� TimeTimeTimeTimeTime k Done ÷ te = WWWWWororororork Done ÷ k Done ÷ te = WWWWWororororork Rk Rk Rk Rk Raaaaate = te = k Done ÷ te = Work rate is the amount of work carried out per unit time. The work completed can be given as a fraction. For example, if only half the task is completed, write 1 2. If the whole task is... |
Megan 6 hours, Margarita 3 hours, and James 4 hours to clean the utensils individually. How long would it take the three of them to clean the utensils if they worked together? 5. A bathtub can be filled from a faucet in 10 minutes. However, a pump can empty the bathtub in 15 minutes. If the faucet and the pump are on ... |
is look through the word problem to identify the work completed and the time taken. Section 2.7 Section 2.7 Section 2.7 — Work-Related Tasks Section 2.7 Section 2.7 123123123123123 TTTTTopicopicopicopicopic 2.7.22.7.2 2.7.22.7.2 2.7.2 California Standards: Students applpplpplpplpplyyyyy Students a Students a 15.0: 15.... |
Combined work rate = combined work completed combined work completed total time total time Here’s the same problem that you saw in Example 2 in Lesson 2.7.1 — but now using the new method. Example Example Example Example Example 11111 An inlet pump can fill a water tank in 10 hours. However, an outlet pump can empty t... |
by by by 12y 12y 12y 12y 12x,,,,, the L Multipl Multipl the L the L CM of CM of 6, 6, 4, 4, and and the LCM of CM of 6, 6, 4, 4, and Multipl Multipl the L CM of 6, 4, and hen solve fe fe fe fe for or or or or x TTTTThen solv hen solv hen solv hen solv x = 2 2 5 Therefore painting the wall would take Jesse and Melinda ... |
3x + 21 = 7x 4x = 21 x = 5 1 4 So Marisa can dig the garden in 5 1 4 hours = 5 hours and 15 minutes. Guided Practice 1. A pump can fill a fuel tank in 30 minutes. A second pump can fill the same tank in 60 minutes. How long would it take to fill the fuel tank if both pumps were filling the tank together? 2. Machine A ... |
them to tile the room if they worked together? 4. A faucet can fill a barrel in 2 hours. However, there is a hole in the bottom of the barrel that can empty it in 6 hours. Without knowing of the hole, Leo tries to fill the barrel. How long will it take? 5. Tyrone, Jerry, and Dorothy are painting a fence surrounding a ... |
check that your solutions make sense in the original word problem. Section 2.7 Section 2.7 Section 2.7 — Work-Related Tasks Section 2.7 Section 2.7 127127127127127 TTTTTopicopicopicopicopic 2.8.12.8.1 2.8.12.8.1 2.8.1 Section 2.8 tions tions alue Equa alue Equa Absolute VVVVValue Equa Absolute Absolute tions alue Equa... |
9 9 –10 –9 –8 –7 –6 –5 –4 –3 –2 – 10 –3 3 The distance between 0 and 3 is the same as the distance between 0 and –3. This can be written as |3| = |–3| = 3. Guided Practice Find the distance that each letter is from zero: B D –5 –4 –3 –2 –. A 3. C Simplify: 5. |–9| 7. –|–2| 2. B 4. D 6. |–20| 8. –|–7| 128128128128128 Se... |
0.02x| = 9 7. |–1.04x| = 0.2392 8. |2x + 1x| = 171 9. |10x – 5x| = 100 + 5 Write these as absolute value equations and find the solutions: 10. The product of four and a number is a distance of 20 from 0. 11. A number has a distance of 8 from 0. 12. Twice a number has a distance of 0.6 from 0. 13. The product of 1.6 and... |
Mor y Inc y Inc alues Ma Absolute VVVVValues Ma alues Ma Absolute Absolute lude More than One y Include Mor alues May Inc e than One lude Mor y Inc alues Ma Absolute Absolute If there’s more than just a single term in the absolute value signs, you need to keep those terms together until the absolute value signs have b... |
of the form d|ax + b| + c = k, rewrite the equation in the form |ax + b| = k − c d, then solve for x. Example Example Example Example Example 22222 Solve 2|3x – 1| + 4 = 12. Solution Solution Solution Solution Solution Rearrange the equation to get the absolute value on its own: 2|3x – 1| + 4 = 12 2|3x – 13x – 1| = 4 ... |
equations rather than two: Example Example Example Example Example 33333 Solve |3x – 2| = |4 – x|. Solution Solution Solution Solution Solution |3x – 2| = |4 – x| ±(3x – 2) = ±(4 – x) There are four possible solutions: (1) 3x – 2 = 4 – x (2) 3x – 2 = –(4 – x) (3) –(3x – 2) = 4 – x (4) –(3x – 2) = –(4 – x) But — this s... |
| = |10 – (2)| |10 – 2| = |8| |8| = |8| 8 = 8 ✓ |5x – 2| = |10 – x| |5(–2) – 2| = |10 – (–2)| |–10 – 2| = |10 + 2| |–12| = |12| 12 = 12 ✓ So x = 2 and x = –2 are the correct solutions of the equation. Guided Practice Find all possible solutions to these absolute value equations: 23. |2x – 8| = |3x – 12| 24. |5x – 7| = ... |
1 2 x + 1 = 5 13. − 8 x. 0 1 ( 10 = 4 ) − x 3 10. − x 4 = 12 12 14 14 In Exercises 15–18 you will need to form an absolute value equation and solve it to find the unknown. 15. If (x + 4) is 3x from 0, what are the possible values of x? 16. If (4x – 5) is (2x + 1) from 0, what are the possible values of x? 17. If (3w +... |
trains are designed to both average 20 mph so that they never meet. However, one train has developed a fault and now travels at 18 mph. The trains set off from stations on opposite sides of the park at 9 a.m. At what time will the faster train catch up with the slower train? (Assume that the trains instantly reach the... |
135135 Chapter 3 Single Variable Linear Inequalities Section 3.1 Inequalities............................................ 137 Section 3.2 Applications of Inequalities................... 147 Section 3.3 Compound Inequalities......................... 155 Section 3.4 Absolute Value Inequalities................... 158 Inve... |
” w Inequalities on the Number Line w Inequalities on the Number Line ou Can Sho YYYYYou Can Sho ou Can Sho w Inequalities on the Number Line ou Can Show Inequalities on the Number Line w Inequalities on the Number Line ou Can Sho The inequality x > 4 represents the interval (part of the number line) where the numbers ... |
x ≥ 1 6. l £ 3 9. 10 £ x 10. State the inequality represented on the number line opposite. 11. State the inequality represented on the number line opposite using interval notation. Independent Practice –4 –2 –4 –2 0 0 2 2 4 4 In Exercises 1–3 write each inequality in interval notation. 1. r > 8 2. t £ –9 3. 3 £ x In E... |
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equalities ty of Inequalities operty of dition Proper Inequalities ty of oper dition Pr Addition Property of Inequalities Given real numbers a, b, and c, if a > b, then a + c > b + c. In other words, adding the same number to both sides of an inequality gives an equivalent inequality. Example Example Example Example Ex... |
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2 ≥ –3 13. 3k + 2 ≥ 2k 15. 2t + 1 £ t – 10 10. t + 5 £ 6 12. y + 3 > –9 14. 4x – 3 < 3x 16. 6(x – 1) ≥ 5(x + 2) Independent Practice Solve the following inequalities. Justify each step and write each solution set in interval notation. 1. j – 3 > –1.5 3. x + 3 < –11 5. 5k – 1.25 < 4k – 4 2. k – 2 ≥ –5 4. 2v – 2.5 > v +... |
uniform. Use this inequality to find the minimum amount of money he needs to earn. 24. An art gallery sells Peter’s paintings for $x, and keeps $100 commission. This means Peter is paid $(x – 100) for each painting. If Peter wants to make at least $750 for a particular painting, write an inequality to represent the am... |
and pr one v one v one v le and pr one v h steppppp..... h ste h ste or eac or eac tion f tion f justifica justifica or each ste tion for eac justification f h ste or eac tion f justifica justifica What it means for you: You’ll solve inequalities that contain × and ÷ signs. Key words: inequality reverse 14214214214214... |
. 1 > x 3 12≤ ≤ y 20 − 1 5 2. 5. 1 4 x 4 p > 12 5> − 8. 1 3 1 < a 9 11. 2 5 p < 4 3. 6 15 9. ≥ k 12 1 4 12. − <6 3 4 a Inequalities Inequalities ty of ty of oper oper vision Pr DiDiDiDiDivision Pr vision Pr Inequalities ty of Inequalities operty of vision Proper Inequalities ty of oper vision Pr Division Property of In... |
. However, if the original inequality is multiplied by a negative number like –3, the resulting inequality is –12 > –9, which is false. To make the resulting inequality true, you have to reverse the inequality sign. That gives –12 < –9, which is true. Multiplication Property of Inequalities — Negative Numbers Given rea... |
and c < 0 then a c b <. c In other words, if you divide both sides of an inequality by a negative number, you have to reverse the inequality symbol — otherwise the statement will be false. Start with an inequality: Dividing by –3 and reversing the inequality sign gives:...is true. –3...which is also true. Example Exam... |
2m + 11 − + x 2 3 1 5 3 − ≥ 16. 17. x 4 11 12 5 9 18. 0.5(x – 1) – 0.75(1 – x) < 0.65(2x – 1) 19. 7 – 3(x – 7) £ 4(x + 5) + 1 20. 0.35(x – 2) – 0.45(x + 1) ≥ 8 + 0.15(x – 10) 21. 3(2x + 6) – 5(x + 8) £ 2x – 22 22. Laura has $5.30 to spend on her lunch. She wants to buy a chicken salad costing $4.20 and decides to spen... |
alities Multistep Inequalities Multistep Inequalities Multistep Inequalities California Standards: 4.0: Students simplify 4.0: Students simplify 4.0: Students simplify 4.0: Students simplify 4.0: Students simplify eeeeexprxprxprxprxpressions bef essions befororororore solving e solving e solving essions bef essions bef... |
hniques Multistep Inequalities Combine Lots of hniques p Inequalities Combine Lots of Multiste Multiste To simplify and therefore solve an inequality in one variable such as x, you need to isolate the terms in x on one side and isolate the numbers on the other. It’s often easiest to keep the x-terms on the side of the... |
ak Complica BrBrBrBrBreak Complica eak Complica eak Complicated Inequalities into Smaller Ste ted Inequalities into Smaller Ste eak Complica Here’s a useful checklist for tackling more complicated inequality questions by breaking them down into easier steps: Solving Inequalities 1. Multiply out any parentheses. 2. Simp... |
3x ≥ 10(x + 1) 23. 2(x – 1) ≥ 4(x – 2) – 8 24. 12(b + 1) – 10b > 7(b + 3) + 6 148148148148148 Section 3.2 Section 3.2 Section 3.2 — Applications of Inequalities Section 3.2 Section 3.2 Independent Practice In Exercises 1–7, solve each inequality. 1. 9x – 7 £ 11 3. 6x – 12 > 5x + 8 2. 8c – 10 ≥ 7c + 6 4. 11(x + 8) ≥ 12... |
Up RRRRRound Up ound Up ound Up ound Up ound Up RRRRRound Up ound Up ound Up If you ever get stuck when you’re solving inequalities with more than one step, refer to the checklist on the previous page. Just take it one step at a time, as if you were dealing with an equation. Section 3.2 Section 3.2 Section 3.2 — Appli... |
justifica justifica What it means for you: You’ll solve real-life problems involving inequalities. Key words: inequality isolate Check it out: These rules are pretty much the same as when you’re dealing with real-life equation problems. Inequalities Inequalities tions of tions of pplica pplica AAAAApplica Inequalities... |
even integer x + 4 = next (third) even integer 150150150150150 Section 3.2 Section 3.2 Section 3.2 — Applications of Inequalities Section 3.2 Section 3.2 Check it out: In this example, you know you need a “>” and not a “≥” because the problem says “more than,” not “more than or equal to.” Example 1 continueduedueduedu... |
perimeter of a rectangular field, F, is given by the formula p = 2l + 2w. As shown in the diagram, the length l = (10x – 6) m and the width w = (5x – 3) m. Find the possible values of x for which the given rectangular field would have a perimeter of at least 1182 meters. (5 – 3) m x F x (10 – 6) m Solution Solution So... |
solve an inequality to find out how long it will be before Car A is traveling faster than Car B. 152152152152152 Section 3.2 Section 3.2 Section 3.2 — Applications of Inequalities Section 3.2 Section 3.2 Example Example Example Example Example 33333 A long-distance telephone call from Los Angeles, California, to Harar... |
T’s checking account has monthly charges of an $8 service fee plus 6¢ per check written. Bank S’s checking account has monthly charges of a $10 service fee plus 4¢ per check written. A company has 150 employees, and pays them monthly by check. The company’s financial adviser suggests that Bank S would be cheaper to us... |
2.4–2.7. Always remember to give your solution as a sentence that answers the original problem. 154154154154154 Section 3.2 Section 3.2 Section 3.2 — Applications of Inequalities Section 3.2 Section 3.2 TTTTTopicopicopicopicopic 3.3.13.3.1 3.3.13.3.1 3.3.1 Section 3.3 Compound Inequalities Compound Inequalities Compou... |
out: The solution set of the conjunction is the numbers for both both both inequalities are which both both true. Sometimes a math problem gives you two different restrictions on a solution, using inequality signs. A compound inequality is two inequalities together — for example, 2x + 1 < 5 and 2x + 1 > –1. lude the W... |
y 2 to get et et et et x in the mid y 2 to g y 2 to g in the middledledledledle vide b vide b in the mid in the mid DiDiDiDiDivide b vide by 2 to g y 2 to g vide b in the mid So the solution is any number greater than –1 but less than 2. This is graphed as: –1 0 1 2 Section 3.3 Section 3.3 Section 3.3 — Compound Inequ... |
x > 8 x > 8 3 2 8 3 DiDiDiDiDivide b vide b vide b vide by 3y 3y 3y 3y 3 vide b In Exercises 20–23, solve the inequality and graph each solution set. 20. 7a – 7 < –7 or 7a – 7 > 21 21. 5x – 4 £ 6 or 5x – 4 ≥ 26 22. c − 9 5 < 3 or c − 9 5 ≥ 9 23. t − 7 3 £ –9 or t − 7 3 > 6 In Exercises 24–27, solve each disjunction. 24... |
three consecutive odd integers is between 155 and 160. Find the consecutive odd integers. The formula C = degrees Celsius. Use this fact to answer Exercises 22–23. (F – 32) is used to convert degrees Fahrenheit to 5 9 22. The temperature inside a greenhouse falls to a minimum of 65 °F at night and rises to a maximum o... |
18 aren’t included. You last saw absolute value equations in Section 2.8 — now you’re going to see inequalities involving absolute values. As with normal inequalities, you can have conjunctions and disjunctions with absolute value inequalities. e Compound Inequalities e Compound Inequalities alue Inequalities ar Absol... |
conjunction rite the inequality as a conjunction e the conjunction to get et et et et x b b b b by itself y itself y itself e the conjunction to g e the conjunction to g SolvSolvSolvSolvSolve the conjunction to g y itself y itself e the conjunction to g Graph: –4 0 18 158158158158158 Section 3.4 Section 3.4 Section 3.... |
3(2 – x)| £ 5 £ 5 1 5 18. 2 x − 5 21. |5x – 11| £ 19 £ 4 19. 3 2 x − 2 £ 4 Check it out: The graphs head in opposite directions — they have no points in common. junctions junctions nequalities CCCCCan an an an an BBBBBe e e e e DisDisDisDisDisjunctions nequalities nequalities alue IIIIInequalities alue alue Absolute VV... |
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