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This ooklet shows you how to find a whole unch of values that fit a given rule! look at a range (more than just one) of possile values. This singer s vocal range is shown on the music stave elow: There are lower and higher notes, ut these ones represent the range this singer can sing. The singer would say: I can sing notes from low C up to and including high A. A mathematician would write: Low C # the notes I can sing # high A. Write down another situation where someone would use a range of values to descrie something. Give this a go! Q This dart oard target has a diameter of 40 cm. Each ring is 0 cm wide. If a dart lands on the line etween two values, the higher value is counted. a Shade the area in which a dart can land in to score 4 points. Write down how you would explain what distances from the outside edge a dart must land on the target to score 4 points. 4 6 4 6 8 6 4 6 4 Work through the ook for a great way to do this I SERIES TOPIC
How does it work? Inequality signs are used when variales can have a range of values. Less than Greater than These two signs mean not including the numer Use inequality signs and variales to replace these statements. (i) The numers (n) that are greater than 4. (ii) All temperatures (t) less than 0 o. n 4 t 0 o Numers greater than does not include 4 Temperatures less than does not include 0 o The numer IS included for these signs. # $ Less than or equal to Greater than or equal to Use inequality signs and variales to replace these statements. (i) All scores (x) of 0 or more. (ii) All values (y) from 0 to. x $ 0 y $ 0 and y # 0 or more includes the score of 0 Both values of 0 and are included y $ 0 is the same as 0 # y 0 # y # comined into a single statement (iii) Any numer (a) smaller than 0 or, greater than or equal to. a 0 and a $ 0 not included, is included These cannot e comined like part (ii) ecause the two inequalities do not overlap. I SERIES TOPIC
How does it work? Your Turn Inequality signs Use inequality signs and variales to replace these statements: a The numer a is less than 3... Any value (y) greater than or equal to... c The temperature (t) is always over 8 o... d A clown lows ules () up to times every party... e All values (n) from - to 3... f The seat numers for row r are etween 30 and 4... g Each vase holds at least flowers (f) ut always less than 0... Join a line etween each statement and the matching inequality for any value x.. Less than or equal to 0 0 x # 0 Between 0 and 40 x # 0, x 40 Numers from 0 to 0 x # 0, x $ 40 Greater than or equal to 0 and smaller than 40 x 0 Numers lower than 40 0 x 40 Numers that are not less than 0 x # 40 Numers less than 0 and greater than 40 x 40 Any value elow 40 and aove 0 x # 0 Greater than or equal to 40 and less than or equal to 0 x 0 INEQUALITY SIGNS $ $ INEQUALITY SIGNS INEQUALITY SIGNS.../.../0... Greater than 0 Greater than 0 up to and including 0 Values under 0 and over 0 Numers greater than 0 Numers that are not greater than 40 Numers not greater than 0 and greater than 40 x 0, x 40 0 # x # 0 x 0, x 0 0 # x 40 x $ 0 0 x 40 I SERIES TOPIC 3
How does it work? Your Turn Inequality signs 3 Match the three values on the left with the correct inequality on the right. Numers can e ordered from least to most using inequalities!,-,-, 0,- 0-0,-, - 0,, 0,-, 0, -, 0, -,- 0,-, -0,-, - -0 0 - - - 0 - # - 0 # 0 0 4 - # - - 0 - - - 0 - # - - 0 4 Join a line etween each statement and the matching inequality for the variales a, and c. You have earned yourself an awesome stamp if you get most of these tricky ones correct!. a is greater than ut less than c a is not greater than c ut greater than a is less than c which is smaller than c is greater than which is not smaller than a c is smaller than and greater than or equal to a is at least a ig as c ut smaller than a c is less than or equal to which is not greater than a c is smaller than a ut not smaller than a # c c # # a a # c a c # c a a c a # c c # a * AWESOME *.../.../0... * AWESOME * 4 I SERIES TOPIC
How does it work? on the numer line are used when variales can have a range of values., #, =, $ Open circle is used for to mean not including this numer Closed circle is used for to mean including this numer The inequality sign tells you which way the arrow should point! Display these inequalities on a numer line. (i) x 0-4 -3 - - 0 3 4 x 0 not included, so use an open circle x 0, same direction of arrow (ii) n # - -4-3 - - 0 3 4 n n # -, same direction of arrow - is included, so use a closed circle (iii) w - 3. -3. is not included -4-3 - - 0 3 4 w -3. plotted half way etween -4 and -3 Simply include a couple of values on either side of the point when laelling. (iv) m $ is included 9 0 3 m Lael axis with correct variale I SERIES TOPIC
How does it work? Your Turn on the numer line Display these inequalities on a numer line: a c d e f g h a $ 4 c -6 d # e -6. f 0. g $ 4 - h -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 a -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8-9 -8-7 -6 - -4-3 - - 0 3 4 6 7 8 c -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 d -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 e -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 f -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 g -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 h Use inequality signs and variales to replace these statements: a -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 a -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 c -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 c d -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 d e -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 e f -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 f g -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 g h -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 h 6 I SERIES TOPIC
How does it work? Your Turn on the numer line 3 Lael these numer lines and display the inequalities. a a # 0 3 4.../.../0... NEQUALITY ON THE NUMBER LINE IINEQUALITY ON THE NUMBER LINE - 0 3 h 9 c x 7 d p -0 e d $ -33 f q #. psst! lael the whole numers efore displaying answer g y 0 - h k # 48 6 i r $ - 3 j h 78 4 I SERIES TOPIC 7
Where does it work? Solving one-step inequalities Just like equations, we use opposite operations to get the variale all y itself. and pairs of opposite operations and Instead of getting just one answer, we get a whole range of answers that work. Solve these one-step inequalities and display the solution on a numer line. (i) a + $ 7 a + - $ 7- a $ All values of or greater will work 0 3 4 a is included in the solution (ii) m -0-8 m - 0 + 0 8 + 0 m All values greater than will work 0 3 4 m is not included in the solution (iii) 4n - 4 4n ' 4-4 ' 4 n - 3. All values less than -3. will work - -4-3 - - n -3. is not included in the solution (iv) #. 3 3 3 #. 3 # # # 7. All values of 7. or less will work 6 7 8 9 7. is included in the solution 8 I SERIES TOPIC
Where does it work? Your Turn Solving one-step inequalities Solve these one-step inequalities and display the solution on a numer line. a x - # 3 4 x.../.../0... INEQUALITIES SOLVING ONE-STEP SOLVING ONE-STEP INEQUALITIES m + 9 - -4-3 - - m c n $ -0-7 -6 - -4-3 n d a 3 # 4 6 7 8 a e + 6. - 6 - -4-3 - f w # 3 3 3 4 6 w g y + 4. 9 - - 0 y I SERIES TOPIC 9
Where does it work? Your Turn Solving one-step inequalities Write a one-step inequality using the given operation sign with a solution matching each displayed answer. a [+] - 0 3 a [-] - -4-3 - - c [#] -7-6 - -4-3 c d ['] 3 4 6 d e [-] -4-3 - - 0 e f [+] - -0-9 -8-7 f g [#] -6 - -4-3 - g h ['] 3 4 6 7 h 0 I SERIES TOPIC
Where does it work? Reversing inequalities Multiplying oth sides y a negative numer means the inequality sign must e reversed (flipped around). Solve these one-step inequalities and display the solution on a numer line. (i) - y y - (- ) (- ) # # y - Reverse the inequality sign -4-3 - - 0 y - is not included in the solution The same rule applies when we divide oth sides y a negative numer. (ii) -g $ - -g '(- ) $ - '(- ) g # 3 Reverse the inequality sign 3 4 g 3 is included in the solution Check this out to see why: If x y x and y are any two numers that fit then x- y y- y Sutract y from oth sides ` x- y 0 then x- x- y 0 - x Sutract x from oth sides ` -y - x The inequality is reversed ` -x - y This is the previous line written in reverse Changing oth sides to negative values is the same as multiplying/dividing y -. I SERIES TOPIC
< REVERSINGREVERSING INEQUALITIES < Where does it work? Your Turn Reversing inequalities Solve these inequalities and display the solution on a numer line. a -x - x - y 6 y c - z $ - z d - 4p # 8 p e -. q - 7. q f - 4 r # 3 r g s ' 9 `- - 7 j s < INEQUALITIES.../.../0... < I SERIES TOPIC
Where does it work? Your Turn Reversing inequalities Match the inequality with the displayed solutions on the numer line to reveal the answer to: The title of the ook y Thomas Harriot (60-6) in which the inequality signs used today were first pulished. a -0x $ - 0 - x 0 c - x $ 4 R 8 9 P -4-3 - - 0 3 4 x -4-3 - - 0 3 4 x -4-3 - - 0 3 4 x - x d 0 e - 8x # 4 A S L I -4-3 - - 0 3 4 x -4-3 - - 0 3 4 x f - 7.x 7 N -4-3 - - 0 3 4 x g - x 0. $ - - x 3 h 9 0 T Y 3 6-4 -3 - - 0 3 4 x -4-3 - - 0 3 4 x - i x X -4-3 - - 0 3 4 x j -x - 3 4-4 -3 - - 0 3 4 x k -.x - 6.6 E C -4-3 - - 0 3 4 x - x 4 l # - 4-4 -3 - - 0 3 4 x 3 6 8 3 0 3 7 6 8 3 4 9 3 8 I SERIES TOPIC 3
Where does it work? Solving two-step inequalities Solve these the exact same way as equations. Use opposite operations to get the variale all y itself. From now on, you only need to display the solutions on a numer line if asked to do so. Solve these two-step inequalities. (i) 3m - 3m - + + 3m 3m 3 3 m 4 Add to oth sides Divide oth sides y 3 All values of m less than 4 will work p (ii) + - 6 (show solution on a numer line) 3 + p # 3-6 # 3 3 + p - 8 -+ p - 8 - Multiply oth sides y 3 Sutract from oth sides p - 3 All values for p greater than -3 will work -4-3 - - -0 p (iii) 9- n # 8 (show solution on a numer line) 9-9- n # 8-9 Sutract 9 from oth sides Coefficient of n is - -n # - -n '(- ) $ - '(- ) Divide oth sides y - n $ All values for n of or greater will work Reverse the inequality when multiplying or dividing y a negative numer - 0 3 n 4 I SERIES TOPIC
Where does it work? Your Turn Solve: Solving two-step inequalities a a + 7 9 3-6 # 30 c + 4c $ 4 d - 7+ d - Solve these inequalities and display the solution on a numer line. a + m 6 # 3 n + 4 3 - p - 6 c. 4-0 + q d $ - SSOLVING TWO-STEP INEQUALITIES 3 Solve: psst: you should add or sutract from oth sides first a a 4 + -6 $ - OLVING TWO-STEP INEQUALITIES.../.../0... I SERIES TOPIC
Where does it work? Your Turn Solving two-step inequalities 4 Solve these inequalities: a - 3w + 4 7 -x -0-4 c -y # - 30 d - -8z $ 4 Earn an awesome passport stamp solving these tricky inequalities containing decimals and fractions. a. a + # 344. 4 + 8 3 4 * AWESOME * * AWESOME *.../.../0... c -97. -3. c $ - 37. 6 d - 37 - d - 6. 8 6 6 I SERIES TOPIC
What else can you do? Doule inequalities Two inequalities can e displayed on the same numer line to show: The values eyond two numers The values etween two numers Display these doule inequalities on the same numer line. (i) w # and w 3. is included 3. is not included -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 w 3. plotted half way etween 3 and 4 (ii) y 7 # - - is not included 7 is included -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 y No arrows for values in etween two points They are useful when needing to find a certain range of answers for doule-sided equations. (iii) # x + 6 # x + and x + 6 # x + 6 - # x + - 6 - # x Solving these two inequalities at the same time Same as normal equations, get the variale y itself Sutract from each part x is all values from to less than is included in the solution is not included in the solution - 0 3 4 6 7 x I SERIES TOPIC 7
What else can you do? Your Turn Doule inequalities Display these doule inequalities on the numer line a s and s 3 t -4 and t $ 0 c u # 4 and u d v -. and v. e -8 w 0 f -4 # x 3-9 -8-7 -6 - -4-3 - - 0 3 4 6 7 8 s -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 t -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 u -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 v -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 w -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 x g 3 # y # 7-9 -8-7 -6 - -4-3 - - 0 3 4 6 7 8 y h - z # 3-9 -8-7 -6 - -4-3 - - 0 3 4 6 7 8 z Write the inequality displayed on these numer lines a -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 h -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 i c -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 j d -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 k e -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 m f -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 n g -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 p h -9-8 -7-6 - -4-3 - - 0 3 4 6 7 8 q 8 I SERIES TOPIC
What else can you do? Your Turn Doule inequalities 3 Solve these doule sided inequalities and display the solution on a numer line. DOUBLE INEQUALITIES.../.../0... DOUBLE INEQUALITIES a x - 3 - # y + # 0 c 3 z # 4 4 d 3 8a 6 e. # +. 4 f -4. # 7c # - g d # 8 I SERIES TOPIC 9
What else can you do? The range of possiilities Playing a straight tennis shot at the net, Raferra must hit the all so that it:. Goes over the tennis net 0.9 m high, and. Lands within.9 m past the tennis net. 0.9 cm.9 m The range of distances for a successful shot at the net are: More than 0.9 m high: Shot height 0.9 m and More than 0 m ut less than or equal to.9 long: 0 m Shot length #.9 m Diagrams can help show the range of possiilities. The third-closest ring to the ullseye in archery scores 7 points. (i) Shade the area the archer can hit to score 7 points on the target. (ii) Each ring is 0 cm wide. What range of distances (d) from the edge of the ullseye must the arrow land to score 7 points if the outer circle is included? 0 cm 40 cm 60 cm ` Arrow must land etween 40 cm and at most, 60 cm away from the ullseye 40 cm distance of arrow from ullseye # 60 cm 40 d # 60 30 40 0 60 70 d (cm) 0 I SERIES TOPIC
POSSIBILITIESPOSSIBILITIESTHE RANGE OF What else can you do? Your Turn The range of possiilities What range of volts (V) can these voltmeters read? a c d 0 0 From volts From volts From volts From volts to volts to volts to volts to volts Use an appropriate scale to display the range of values each voltmeter can read on these numer lines: a 0 0 0 V 8 0 4 6 8 V c d 0 00 00 300 400 00 V 9 3 7 9 V 3 The photo shows a competitor jumping into third place in the high jump. The top five jumps made efore the photographed jump was made are listed elow: 8 cm 80 cm 78 cm 76 cm 74 cm a What heights (h) must this competitor have jumped aove/elow to finish clear third? Aove cm and elow cm THE RANGE OF Write these heights as an inequality: h.../.../0... c Display the range of possiilities on the numer line elow using intervals of cm. h (cm) I SERIES TOPIC
What else can you do? Your Turn The range of possiilities 4 A TV chef says during one show: After cooking for 8 minutes a steak is medium. There is usually a difference of 3 minutes cooking time for a steak eing rare or medium and medium or well done. a Calculate and display the range of times (t) this chef would expect it to take for a steak e cooked: (i) Rare? (ii) Well done? # t t # 4 6 7 8 t (min) 7 8 9 0 t (min) If a steak is cooked longer than the time needed for it to e called well done it is considered overdone. Display the inequality showing the cooking times that will make a steak overdone. 9 0 3 t (min) This dart oard target has a diameter of 40 cm. Each ring is 0 cm wide. If a dart lands on the line etween two values, the higher value is counted. Rememer me? a Shade the area a dart can land in to score 4 points. Write and display the inequality on a numer line showing the distances (d) from the outside edge of the target that a dart will score 4 points. 4 6 4 6 8 6 4 # d 0 0 0 30 40 d (cm) 6 4 c Write and display on the same numer plane, the two pairs of inequalities that show the distances from the outside edge to the centre of the target where a dart will not score 4. # d and # d # 0 0 0 d (cm) d Explain in one sentence why the numer lines in this question do not have arrows on the ends. I SERIES TOPIC
What else can you do? Your Turn The range of possiilities 6 In a game of ten pin owling, what range of scores (s) can Janet achieve on her first turn? a # s # Why wasn t the word etween used for the range of scores in this question? 7 One ream of paper contains 00 sheets. A printer cartridge can last for anywhere from 00 up to 00 sheets of printing. Display the range of paper reams (r) one cartridge can e expected to last for? # r # r (reams of paper) 8 A girl reads approximately 600 to 000 words per minute. What numer of pages (n) does she usually read in minutes from a ook that averages 00 words per page? # n # n (pages read) 9 Mizuki is playing a slam shot at the centre of the net in a game of volleyall. To score a point, the all must hit the ground inside the court (or on the line), on the other side of the net..4 cm 9 m 9 m a Write down all the inequalities for the dimensions (height, length and width) that she needs to consider when making this a good shot. The centre of the net = 0 m for width. Write down all the inequalities for the dimensions (d) of a shot that will result in Mizuki losing a point. I SERIES TOPIC 3
Cheat Sheet Here is a summary of the things you need to rememer for Inequality signs Less than Greater than value given not included # $ Less than or equal to value given included Greater than or equal to on the numer line Open circle Closed circle # $ Less than Greater than Less than or equal to Greater than or equal to Solving one-step and two-step inequalities Just like for equations, we use opposite operations to get the variale all y itself. and pairs of opposite operations and Reversing inequalities The inequality sign must e reversed (flipped around) when multiplying/dividing oth sides y a negative numer. ` If x y then -x -y (this rule applies for all inequality signs:,, #, $) Doule inequalities Two inequalities can e displayed on the same numer line to show: The values eyond two numers The values etween two numers 4 I SERIES TOPIC
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