Algebra - Solving Inequalities
Question 1
Solve, showing your solution on a number line
a) $x-1>4$
a
$x>5$
Question ID: 20200010010
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b) $x-3<5$
a
$x<8$
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c) $x-7>9$
a
$x>16$
Question ID: 20200010030
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d) $x-10\le20$
a
$x\le30$
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e) $x-50<2$
a
$x<52$
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f) $x-78\ge13$
a
$x\ge91$
Question ID: 20200010060
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Question 2
Solve, showing your solution on a number line
a) $x+2<7$
a
$x<5$
Question ID: 20200020010
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b) $x+5\ge11$
a
$x\ge6$
Question ID: 20200020020
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c) $x+10\le50$
a
$x\le40$
Question ID: 20200020030
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d) $x+12>33$
a
$x>21$
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e) $x+25\ge140$
a
$x\ge115$
Question ID: 20200020050
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f) $x+19>100$
a
$x>81$
Question ID: 20200020060
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Question 3
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a) $2x>10$
a
$x>5$
Question ID: 20200030010
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b) $6x>18$
a
$x>3$
Question ID: 20200030020
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c) $10x\le80$
a
$x\le8$
Question ID: 20200030030
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d) $6x\ge36$
a
$x\ge6$
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e) $7x<56$
a
$x<8$
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f) $9x\le72$
a
$x\le8$
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Question 4
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a) $\frac x3\ge2$
a
$x\ge6$
Question ID: 20200040010
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b) $\frac x5\le5$
a
$x\le25$
Question ID: 20200040020
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c) $\frac x9\le3$
a
$x\le27$
Question ID: 20200040030
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d) $\frac x4>10$
a
$x>40$
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e) $\frac {x}{11}<5$
a
$x<55$
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f) $\frac {x}{12}\ge6$
a
$x\ge72$
Question ID: 20200040060
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Question 5
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a) $-2x<4$
a
$x>-2$
Question ID: 20200050010
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b) $-5x>10$
a
$x<-2$
Question ID: 20200050020
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c) $-3x\ge18$
a
$x\le-6$
Question ID: 20200050030
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d) $-8x\le64$
a
$x\ge-8$
Question ID: 20200050040
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e) $-9x\ge99$
a
$x\le-11$
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f) $-x<13$
a
$x>-13$
Question ID: 20200050060
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Question 6
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a) $\frac {x}{-2}>3$
a
$x<-6$
Question ID: 20200060010
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b) $-\frac {x}{4}\le5$
a
$x\ge-20$
Question ID: 20200060020
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c) $\frac {x}{-6}\ge9$
a
$x\le-54$
Question ID: 20200060030
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d) $-\frac {x}{7}>11$
a
$x<-77$
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e) $-\frac {x}{3}<7$
a
$x>-21$
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f) $\frac {x}{-9}\le12$
a
$x\ge-108$
Question ID: 20200060060
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Question 7
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a) $2x+3<13$
a
$x<5$
Question ID: 20200070010
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b) $4x-2\ge30$
a
$x\ge8$
Question ID: 20200070020
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c) $\frac x3+4\le10$
a
$x\le2$
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d) $\frac x7-3\le3$
a
$x\le42$
Question ID: 20200070040
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e) $9x-13\ge5$
a
$x\ge2$
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f) $\frac x9+17>25$
a
$x>72$
Question ID: 20200070060
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Question 8
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a) $12-2x>18$
a
$x<-3$
Question ID: 20200080010
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b) $58-6x\ge100$
a
$x\le-7$
Question ID: 20200080020
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c) $\frac{x}{-5}+16\ge19$
a
$x\le-15$
Question ID: 20200080030
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d) $9-\frac{x}{2}\le15$
a
$x\ge-12$
Question ID: 20200080040
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e) $90-10x<200$
a
$x>-11$
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f) $311-\frac x8<320$
a
$x>-72$
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Question 9
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a) $2(x+3)<14$
a
$x<4$
Question ID: 20200090010
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b) $3(x-2)\ge27$
a
$x\ge11$
Question ID: 20200090020
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c) $-5(x+3)\le-20$
a
$x\ge1$
Question ID: 20200090030
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d) $-7(x-5)>-63$
a
$x<14$
Question ID: 20200090040
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e) $6(x+5)<54$
a
$x<4$
Question ID: 20200090050
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f) $-20(x-1)\le-60$
a
$x\ge4$
Question ID: 20200090060
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Question 10
Solve, showing your solution on a number line
a) $\frac{x+2}{5}>3$
a
$x>13$
Question ID: 20200100010
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b) $\frac{x-3}{6}>4$
a
$x>27$
Question ID: 20200100020
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c) $\frac{x+7}{2}\le8$
a
$x\le9$
Question ID: 20200100030
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d) $-\frac{x+1}{7}>-10$
a
$x<69$
Question ID: 20200100040
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e) $\frac{x-3}{-8}\ge-6$
a
$x\le51$
Question ID: 20200100050
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f) $-\frac{x-10}{9}\ge-11$
a
$x\le109$
Question ID: 20200100060
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Question 11
Solve, showing your solution on a number line
a) $4(8-x)<20$
a
$x>3$
Question ID: 20200110010
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b) $\frac{13-x}{6}\le2$
a
$x\ge1$
Question ID: 20200110020
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c) $7(10-x)\ge35$
a
$x\le5$
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d) $\frac{80-x}{8}\le8$
a
$x\ge16$
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e) $-3(9-x)>-30$
a
$x>-1$
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f) $-\frac{17-x}{3}<4$
a
$x<5$
Question ID: 20200110060
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Question 12
Solve, showing your solution on a number line
a) $4\le x+3\le5$
a
$1\le x\le2$
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b) $7\le x-2\le10$
a
$9\le x\le12$
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c) $10\le2x\le20$
a
$5\le x\le10$
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d) $5\le\frac x2<8$
a
$10\le x<16$
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e) $19\le2x+3\le25$
a
$8\le x\le11$
Question ID: 20200120050
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f) $4\le5x-1<19$
a
$1\le x<4$
Question ID: 20200120060
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Question 13
Solve, showing your solution on a number line
a) $-10\le-2x\le-6$
a
$3\le x\le5$
Question ID: 20200130010
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b) $-7\le\frac{x}{-3}\le-5$
a
$15\le x\le21$
Question ID: 20200130020
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c) $-13\le3-2x\le1$
a
$1\le x\le5$
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d) $3\le3-\frac{x}{5}\le8$
a
$-25\le x\le0$
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e) $-30\le-5(x-4)\le-10$
a
$6\le x\le10$
Question ID: 20200130050
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f) $-10\le\frac{5-5x}{6}<-5$
a
$7 < x\le13$
Question ID: 20200130060
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$-10\le\frac{5-5x}{6}<-5$
$-60\le5-5x<-30$
$-65\le-5x<-35$
$13\ge x>7$
$7 < x\le13$
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Question 14
By forming an inequation, solve the following
a) I ate $5$ more chocolates than Harry. We lost count of how many chocolates we ate, but it was certainly more than $21$! What is the possible number of chocolates Harry ate?
a
Harry ate more than $8$ chocolates
Question ID: 20200140010
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I ate $5$ more chocolates than Harry. We lost count of how many chocolates we ate, but it was certainly more than $21$! What is the possible number of chocolates Harry ate?
Let the number of chocolates I ate be $c$ and the number of chocolates Harry ate be $h$
Then
$c+h>21$
Also as I ate $5$ more chocolates:
$c=h+5$
Therefore
$\begin{align}h+5+h&>21\\2h+5&>21\\2h&>16\\h&>8\end{align}$
Harry ate more than $8$ chocolates
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b) Joe spent twice as much as Jenna on a shopping trip yesterday. They know they spent less than £$450$. How much could Jenna have spent?
a
Jenna spent less than £$150$
Question ID: 20200140020
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Joe spent twice as much as Jenna on a shopping trip yesterday. They know they spent less than £$450$. How much could Jenna have spent?
Let the amount Joe spent be $a$ and the amount Jenna spent be $b$
Then
$a+b<450$
Also, Jenna spent twice as much as Joe, so:
$a=2b$
Therefore
$\begin{align}2b+b&<450\\3b&<450\\b&<150\end{align}$
Jenna spent less than £$150$
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c) The average of $3,6,7$ and $x$ is greater than or equal to $5$. What range of numbers could $x$ be?
a
$x$ must be greater than or equal to $4$
Question ID: 20200140030
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The average of $3,6,7$ and $x$ is greater than or equal to $5$. What range of numbers could $x$ be?
$\begin{align}\frac{3+6+7+x}{4}&\ge5\\\frac{16+x}{4}&\ge5\\16+x&\ge20\\x&\ge4\end{align}$
$x$ must be greater than or equal to $4$
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d) The speed of a ball thrown down from a building is given as $s=3+9t$, where $s$ is the speed at which the ball is travelling in $m/s$ and $t$ is the time in seconds after the ball is thrown. At what times is the speed between $21m/s$ and $39m/s$?
a
The ball is travelling between $21m/s$ and $39m/s$ after $2$ seconds and before $4$ seconds
Question ID: 20200140040
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The speed of a ball thrown down from a building is given as $s=3+9t$, where $s$ is the speed at which the ball is travelling in $m/s$ and $t$ is the time in seconds after the ball is thrown. At what times is the speed between $21m/s$ and $39m/s$?
Form a double inequality:
$21<3+9t<39$
$21-3<9t<39-3$
$18<9t<36$
$2 < t<4$
The ball is travelling between $21m/s$ and $39m/s$ after $2$ seconds and before $4$ seconds
Question ID: 20200140040
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Question 1
Solve, showing your solution on a number line
a) $x-1>4$
b) $x-3<5$
c) $x-7>9$
d) $x-10\le20$
e) $x-50<2$
f) $x-78\ge13$
Question 2
Solve, showing your solution on a number line
a) $x+2<7$
b) $x+5\ge11$
c) $x+10\le50$
d) $x+12>33$
e) $x+25\ge140$
f) $x+19>100$
Question 3
Solve, showing your solution on a number line
a) $2x>10$
b) $6x>18$
c) $10x\le80$
d) $6x\ge36$
e) $7x<56$
f) $9x\le72$
Question 4
Solve, showing your solution on a number line
a) $\frac x3\ge2$
b) $\frac x5\le5$
c) $\frac x9\le3$
d) $\frac x4>10$
e) $\frac {x}{11}<5$
f) $\frac {x}{12}\ge6$
Question 5
Solve, showing your solution on a number line
a) $-2x<4$
b) $-5x>10$
c) $-3x\ge18$
d) $-8x\le64$
e) $-9x\ge99$
f) $-x<13$
Question 6
Solve, showing your solution on a number line
a) $\frac {x}{-2}>3$
b) $-\frac {x}{4}\le5$
c) $\frac {x}{-6}\ge9$
d) $-\frac {x}{7}>11$
e) $-\frac {x}{3}<7$
f) $\frac {x}{-9}\le12$
Question 7
Solve, showing your solution on a number line
a) $2x+3<13$
b) $4x-2\ge30$
c) $\frac x3+4\le10$
d) $\frac x7-3\le3$
e) $9x-13\ge5$
f) $\frac x9+17>25$
Question 8
Solve, showing your solution on a number line
a) $12-2x>18$
b) $58-6x\ge100$
c) $\frac{x}{-5}+16\ge19$
d) $9-\frac{x}{2}\le15$
e) $90-10x<200$
f) $311-\frac x8<320$
Question 9
Solve, showing your solution on a number line
a) $2(x+3)<14$
b) $3(x-2)\ge27$
c) $-5(x+3)\le-20$
d) $-7(x-5)>-63$
e) $6(x+5)<54$
f) $-20(x-1)\le-60$
Question 10
Solve, showing your solution on a number line
a) $\frac{x+2}{5}>3$
b) $\frac{x-3}{6}>4$
c) $\frac{x+7}{2}\le8$
d) $-\frac{x+1}{7}>-10$
e) $\frac{x-3}{-8}\ge-6$
f) $-\frac{x-10}{9}\ge-11$
Question 11
Solve, showing your solution on a number line
a) $4(8-x)<20$
b) $\frac{13-x}{6}\le2$
c) $7(10-x)\ge35$
d) $\frac{80-x}{8}\le8$
e) $-3(9-x)>-30$
f) $-\frac{17-x}{3}<4$
Question 12
Solve, showing your solution on a number line
a) $4\le x+3\le5$
b) $7\le x-2\le10$
c) $10\le2x\le20$
d) $5\le\frac x2<8$
e) $19\le2x+3\le25$
f) $4\le5x-1<19$
Question 13
Solve, showing your solution on a number line
a) $-10\le-2x\le-6$
b) $-7\le\frac{x}{-3}\le-5$
c) $-13\le3-2x\le1$
d) $3\le3-\frac{x}{5}\le8$
e) $-30\le-5(x-4)\le-10$
f) $-10\le\frac{5-5x}{6}<-5$
Question 14
By forming an inequation, solve the following
a) I ate $5$ more chocolates than Harry. We lost count of how many chocolates we ate, but it was certainly more than $21$! What is the possible number of chocolates Harry ate?
b) Joe spent twice as much as Jenna on a shopping trip yesterday. They know they spent less than £$450$. How much could Jenna have spent?
c) The average of $3,6,7$ and $x$ is greater than or equal to $5$. What range of numbers could $x$ be?
d) The speed of a ball thrown down from a building is given as $s=3+9t$, where $s$ is the speed at which the ball is travelling in $m/s$ and $t$ is the time in seconds after the ball is thrown. At what times is the speed between $21m/s$ and $39m/s$?
Answers
Question 1
a) $x>5$

b) $x<8$

c) $x>16$

d) $x\le30$

e) $x<52$

f) $x\ge91$

Question 2
a) $x<5$
b) $x\ge6$
c) $x\le40$
d) $x>21$
e) $x\ge115$
f) $x>81$
Question 3
a) $x>5$
b) $x>3$
c) $x\le8$
d) $x\ge6$
e) $x<8$
f) $x\le8$
Question 4
a) $x\ge6$
b) $x\le25$
c) $x\le27$
d) $x>40$
e) $x<55$
f) $x\ge72$
Question 5
a) $x>-2$
b) $x<-2$
c) $x\le-6$
d) $x\ge-8$
e) $x\le-11$
f) $x>-13$
Question 6
a) $x<-6$
b) $x\ge-20$
c) $x\le-54$
d) $x<-77$
e) $x>-21$
f) $x\ge-108$
Question 7
a) $x<5$

b) $x\ge8$

c) $x\le2$

d) $x\le42$

e) $x\ge2$

f) $x>72$

Question 8
a) $x<-3$
b) $x\le-7$
c) $x\le-15$
d) $x\ge-12$
e) $x>-11$
f) $x>-72$
Question 9
a) $x<4$
b) $x\ge11$
c) $x\ge1$
d) $x<14$
e) $x<4$
f) $x\ge4$
Question 10
a) $x>13$
b) $x>27$
c) $x\le9$
d) $x<69$
e) $x\le51$
f) $x\le109$
Question 11
a) $x>3$
b) $x\ge1$
c) $x\le5$
d) $x\ge16$
e) $x>-1$
f) $x<5$
Question 12
a) $1\le x\le2$

b) $9\le x\le12$

c) $5\le x\le10$

d) $10\le x<16$

e) $8\le x\le11$

f) $1\le x<4$

Question 13
a) $3\le x\le5$
b) $15\le x\le21$
c) $1\le x\le5$
d) $-25\le x\le0$
e) $6\le x\le10$
f) $7 < x\le13$
Question 14
a) Harry ate more than $8$ chocolates
b) Jenna spent less than £$150$
c) $x$ must be greater than or equal to $4$
d) The ball is travelling between $21m/s$ and $39m/s$ after $2$ seconds and before $4$ seconds
Question 1
Solve, showing your solution on a number line
a) $x-1>4$
b) $x-3<5$
c) $x-7>9$
d) $x-10\le20$
e) $x-50<2$
f) $x-78\ge13$
Question 2
Solve, showing your solution on a number line
a) $x+2<7$
b) $x+5\ge11$
c) $x+10\le50$
d) $x+12>33$
e) $x+25\ge140$
f) $x+19>100$
Question 3
Solve, showing your solution on a number line
a) $2x>10$
b) $6x>18$
c) $10x\le80$
d) $6x\ge36$
e) $7x<56$
f) $9x\le72$
Question 4
Solve, showing your solution on a number line
a) $\frac x3\ge2$
b) $\frac x5\le5$
c) $\frac x9\le3$
d) $\frac x4>10$
e) $\frac {x}{11}<5$
f) $\frac {x}{12}\ge6$
Question 5
Solve, showing your solution on a number line
a) $-2x<4$
b) $-5x>10$
c) $-3x\ge18$
d) $-8x\le64$
e) $-9x\ge99$
f) $-x<13$
Question 6
Solve, showing your solution on a number line
a) $\frac {x}{-2}>3$
b) $-\frac {x}{4}\le5$
c) $\frac {x}{-6}\ge9$
d) $-\frac {x}{7}>11$
e) $-\frac {x}{3}<7$
f) $\frac {x}{-9}\le12$
Question 7
Solve, showing your solution on a number line
a) $2x+3<13$
b) $4x-2\ge30$
c) $\frac x3+4\le10$
d) $\frac x7-3\le3$
e) $9x-13\ge5$
f) $\frac x9+17>25$
Question 8
Solve, showing your solution on a number line
a) $12-2x>18$
b) $58-6x\ge100$
c) $\frac{x}{-5}+16\ge19$
d) $9-\frac{x}{2}\le15$
e) $90-10x<200$
f) $311-\frac x8<320$
Question 9
Solve, showing your solution on a number line
a) $2(x+3)<14$
b) $3(x-2)\ge27$
c) $-5(x+3)\le-20$
d) $-7(x-5)>-63$
e) $6(x+5)<54$
f) $-20(x-1)\le-60$
Question 10
Solve, showing your solution on a number line
a) $\frac{x+2}{5}>3$
b) $\frac{x-3}{6}>4$
c) $\frac{x+7}{2}\le8$
d) $-\frac{x+1}{7}>-10$
e) $\frac{x-3}{-8}\ge-6$
f) $-\frac{x-10}{9}\ge-11$
Question 11
Solve, showing your solution on a number line
a) $4(8-x)<20$
b) $\frac{13-x}{6}\le2$
c) $7(10-x)\ge35$
d) $\frac{80-x}{8}\le8$
e) $-3(9-x)>-30$
f) $-\frac{17-x}{3}<4$
Question 12
Solve, showing your solution on a number line
a) $4\le x+3\le5$
b) $7\le x-2\le10$
c) $10\le2x\le20$
d) $5\le\frac x2<8$
e) $19\le2x+3\le25$
f) $4\le5x-1<19$
Question 13
Solve, showing your solution on a number line
a) $-10\le-2x\le-6$
b) $-7\le\frac{x}{-3}\le-5$
c) $-13\le3-2x\le1$
d) $3\le3-\frac{x}{5}\le8$
e) $-30\le-5(x-4)\le-10$
f) $-10\le\frac{5-5x}{6}<-5$
Question 14
By forming an inequation, solve the following
a) I ate $5$ more chocolates than Harry. We lost count of how many chocolates we ate, but it was certainly more than $21$! What is the possible number of chocolates Harry ate?
b) Joe spent twice as much as Jenna on a shopping trip yesterday. They know they spent less than £$450$. How much could Jenna have spent?
c) The average of $3,6,7$ and $x$ is greater than or equal to $5$. What range of numbers could $x$ be?
d) The speed of a ball thrown down from a building is given as $s=3+9t$, where $s$ is the speed at which the ball is travelling in $m/s$ and $t$ is the time in seconds after the ball is thrown. At what times is the speed between $21m/s$ and $39m/s$?
Answers
Question 1
a) $x>5$

b) $x<8$

c) $x>16$

d) $x\le30$

e) $x<52$

f) $x\ge91$

Question 2
a) $x<5$
b) $x\ge6$
c) $x\le40$
d) $x>21$
e) $x\ge115$
f) $x>81$
Question 3
a) $x>5$
b) $x>3$
c) $x\le8$
d) $x\ge6$
e) $x<8$
f) $x\le8$
Question 4
a) $x\ge6$
b) $x\le25$
c) $x\le27$
d) $x>40$
e) $x<55$
f) $x\ge72$
Question 5
a) $x>-2$
b) $x<-2$
c) $x\le-6$
d) $x\ge-8$
e) $x\le-11$
f) $x>-13$
Question 6
a) $x<-6$
b) $x\ge-20$
c) $x\le-54$
d) $x<-77$
e) $x>-21$
f) $x\ge-108$
Question 7
a) $x<5$

b) $x\ge8$

c) $x\le2$

d) $x\le42$

e) $x\ge2$

f) $x>72$

Question 8
a) $x<-3$
b) $x\le-7$
c) $x\le-15$
d) $x\ge-12$
e) $x>-11$
f) $x>-72$
Question 9
a) $x<4$
b) $x\ge11$
c) $x\ge1$
d) $x<14$
e) $x<4$
f) $x\ge4$
Question 10
a) $x>13$
b) $x>27$
c) $x\le9$
d) $x<69$
e) $x\le51$
f) $x\le109$
Question 11
a) $x>3$
b) $x\ge1$
c) $x\le5$
d) $x\ge16$
e) $x>-1$
f) $x<5$
Question 12
a) $1\le x\le2$

b) $9\le x\le12$

c) $5\le x\le10$

d) $10\le x<16$

e) $8\le x\le11$

f) $1\le x<4$

Question 13
a) $3\le x\le5$
b) $15\le x\le21$
c) $1\le x\le5$
d) $-25\le x\le0$
e) $6\le x\le10$
f) $7 < x\le13$
Question 14
a) Harry ate more than $8$ chocolates
b) Jenna spent less than £$150$
c) $x$ must be greater than or equal to $4$
d) The ball is travelling between $21m/s$ and $39m/s$ after $2$ seconds and before $4$ seconds
Solutions
Question 1
a)
b)
c)
d)
e)
f)
Question 2
a)
b)
c)
d)
e)
f)
Question 3
a)
b)
c)
d)
e)
f)
Question 4
a)
b)
c)
d)
e)
f)
Question 5
a)
b)
c)
d)
e)
f)
Question 6
a)
b)
c)
d)
e)
f)
Question 7
a)
b)
c)
d)
e)
f)
Question 8
a)
b)
c)
d)
e)
f)
Question 9
a)
b)
c)
d)
e)
f)
Question 10
a)
b)
c)
d)
e)
f)
Question 11
a)
b)
c)
d)
e)
f)
Question 12
a)
b)
c)
d)
e)
f)
Question 13
a)
b)
c)
d)
e)
f) $-10\le\frac{5-5x}{6}<-5$
$-60\le5-5x<-30$
$-65\le-5x<-35$
$13\ge x>7$
$7 < x\le13$
Question 14
a) I ate $5$ more chocolates than Harry. We lost count of how many chocolates we ate, but it was certainly more than $21$! What is the possible number of chocolates Harry ate?
Let the number of chocolates I ate be $c$ and the number of chocolates Harry ate be $h$
Then
$c+h>21$
Also as I ate $5$ more chocolates:
$c=h+5$
Therefore
$\begin{align}h+5+h&>21\\2h+5&>21\\2h&>16\\h&>8\end{align}$
Harry ate more than $8$ chocolates
b) Joe spent twice as much as Jenna on a shopping trip yesterday. They know they spent less than £$450$. How much could Jenna have spent?
Let the amount Joe spent be $a$ and the amount Jenna spent be $b$
Then
$a+b<450$
Also, Jenna spent twice as much as Joe, so:
$a=2b$
Therefore
$\begin{align}2b+b&<450\\3b&<450\\b&<150\end{align}$
Jenna spent less than £$150$
c) The average of $3,6,7$ and $x$ is greater than or equal to $5$. What range of numbers could $x$ be?
$\begin{align}\frac{3+6+7+x}{4}&\ge5\\\frac{16+x}{4}&\ge5\\16+x&\ge20\\x&\ge4\end{align}$
$x$ must be greater than or equal to $4$
d) The speed of a ball thrown down from a building is given as $s=3+9t$, where $s$ is the speed at which the ball is travelling in $m/s$ and $t$ is the time in seconds after the ball is thrown. At what times is the speed between $21m/s$ and $39m/s$?
Form a double inequality:
$21<3+9t<39$
$21-3<9t<39-3$
$18<9t<36$
$2 < t<4$
The ball is travelling between $21m/s$ and $39m/s$ after $2$ seconds and before $4$ seconds