Algebra - Solving Inequalities with Unknowns on Both Sides
Question 1
Solve the following, showing your solution on a number line
a) $2x>x+3$
a
$x>3$
Question ID: 20230010010
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$\begin{align}2x&>x+3\\2x-x&>3\\x&>3\end{align}$
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b) $6x>5x+7$
a
$x>7$
Question ID: 20230010020
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c) $5+10x\le11x$
a
$5\le x$
or
$x\ge5$
Question ID: 20230010030
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$\begin{align}5+10x&\le11x\\5&\le11x-10x\\5&\le x\\x&\ge5\end{align}$
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d) $29x+10\ge30x$
a
$10\ge x$
or
$x\le10$
Question ID: 20230010040
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e) $12y-4>13y$
a
$-4>y$
or
$y<-4$
Question ID: 20230010050
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f) $-6z<16-7z$
a
$z<16$
Question ID: 20230010060
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Question 2
Solve the following, showing your solution on a number line
a) $3x\ge x+8$
a
$x\ge4$
Question ID: 20230020010
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$\begin{align}3x&\ge x+8\\3x-x&\ge8\\2x&\ge8\\x&\ge8\div2\\x&\ge4\end{align}$
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b) $11x\ge4x-42$
a
$x\ge-6$
Question ID: 20230020020
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$\begin{align}11x&\ge4x-42\\11x-4x&\ge-42\\7x&\ge-42\\x&\ge-42\div7\\x&\ge-6\end{align}$
Question ID: 20230020020
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c) $36-2x>4x$
a
$6>x$
or
$x<6$
Question ID: 20230020030
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d) $4m<19-15m$
a
$x<1$
Question ID: 20230020040
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e) $22-x\le 43x$
a
$\frac12\le x$
or
$x\ge\frac12$
Question ID: 20230020050
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f) $11a\le -100-9a$
a
$x\le-5$
Question ID: 20230020060
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Question 3
Solve the following, showing your solution on a number line
a) $\frac{x+6}{3}\le x$
a
$x\ge3$
Question ID: 20230025010
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$\begin{align}\frac{x+6}{3}&\le x\\x+2&\le3x\\2&\le3x-x\\6&\le2x\\3&\le x\\x&\ge3\end{align}$
Question ID: 20230025010
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b) $\frac{30-4x}{2}\ge x$
a
$x\le5$
Question ID: 20230025020
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$\begin{align}\frac{18-4x}{2}&\ge x\\18-4x&\ge2x\\18&\ge6x\\3&\ge x\\x&\le3\end{align}$
Question ID: 20230025020
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c) $-12x> \frac{4x+100}{3}$
a
$x<-\frac52$
Question ID: 20230025030
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$\begin{align}-12x&> \frac{4x+100}{3}\\-36x&>4x+100\\-40x&>100\\x&<\frac{100}{-40}\\x&<-\frac52\\\end{align}$
Question ID: 20230025030
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Question 4
Solve the following, showing your solution on a number line
a) $x+4\le2x+3$
a
$1\le x$
or
$x\ge1$
Question ID: 20230030010
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b) $5x+7\ge4x+10$
a
$x\ge3$
Question ID: 20230030020
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c) $18x-5<17x+80$
a
$x<85$
Question ID: 20230030030
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d) $4-6x>9-7x$
a
$x>5$
Question ID: 20230030040
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e) $6-4a<8-3a$
a
$-2 < a$
or
$a>-2$
Question ID: 20230030050
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f) $-12-16h<-17h-44$
a
$h<-32$
Question ID: 20230030060
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Question 5
Solve the following, showing your solution on a number line
a) $15+5x\ge3x+25$
a
$x\ge5$
Question ID: 20230040010
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b) $20+7x\le10x+17$
a
$1\le x$
or
$x\ge1$
Question ID: 20230040020
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c) $12x-3<19+20x$
a
$2 < x$
or
$x>2$
Question ID: 20230040030
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d) $2w-25>8-9w$
a
$w>3$
Question ID: 20230040040
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e) $16-7n\le10-10n$
a
$n\le-2$
Question ID: 20230040050
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f) $8x+20>-9x-31$
a
$x>-3$
Question ID: 20230040060
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Question 6
Solve the following, showing your solution on a number line
a) $2(x+4)\ge x+10$
a
$x\ge2$
Question ID: 20230050010
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b) $5(x+3)<4x+55$
a
$x<40$
Question ID: 20230050020
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c) $3(x-4)<2x$
a
$x<12$
Question ID: 20230050030
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d) $4+10x\ge9(x+3)$
a
$x\ge23$
Question ID: 20230050040
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e) $16-4x>5(4-x)$
a
$x>4$
Question ID: 20230050050
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f) $-8(x+6)\le20-7x$
a
$-68\le x$
or
$x\ge-68$
Question ID: 20230050060
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Question 7
Solve the following, showing your solution on a number line
a) $5(x+1)>2x+14$
a
$x>3$
Question ID: 20230060010
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b) $6x+44>2(x+30)$
a
$x>4$
Question ID: 20230060020
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c) $3(x-3)\le11+5x$
a
$-10\le x$
or
$x\ge-10$
Question ID: 20230060030
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d) $3(2x+10)\ge15x+3$
a
$3\ge x$
or
$x\le3$
Question ID: 20230060040
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e) $5-7x\le-2(3-2x)$
a
$1\le x$
or
$x\ge1$
Question ID: 20230060050
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f) $-5(12-4x)\ge-10x+90$
a
$x\ge5$
Question ID: 20230060060
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Question 8
Solve the following, showing your solution on a number line
a) $4(x+6)>5(x+2)$
a
$14>x$
or
$x<14$
Question ID: 20230070010
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b) $4(2x+3)\le7(x+5)$
a
$x\le23$
Question ID: 20230070020
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c) $7(3x-3)>11(2x-4)$
a
$23>x$
or
$x<23$
Question ID: 20230070030
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d) $6(4+x)\le-5(2-x)$
a
$x\le-34$
Question ID: 20230070040
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e) $-5(6-5x)>-8(-3x-5)$
a
$x>70$
Question ID: 20230070050
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f) $-9(7-9x)\le4(20x-16)$
a
$x\le-1$
Question ID: 20230070060
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Question 9
Solve the following, showing your solution on a number line
a) $6(x+4)\ge4(x+9)$
a
$x\ge6$
Question ID: 20230080010
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b) $10(x+6)\ge15(x+4)$
a
$0\ge x$
or
$x\le0$
Question ID: 20230080020
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c) $8(x-4)<14(x+5)$
a
$-17>x$
or
$x<-17$
Question ID: 20230080030
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d) $4(3x+2)>-3(4+x)$
a
$x>-\frac34$
Question ID: 20230080040
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e) $5(5-4x)<5(2x-1)$
a
$x>1$
Question ID: 20230080050
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f) $-5(2-x)\ge7(5+5x)$
a
$-\frac32\ge x$
or
$x\le-\frac32$
Question ID: 20230080060
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Question 10
Solve the following, showing your solution on a number line
a) $2x+1> \frac{3x+75}{5}$
a
$x>10$
Question ID: 20230085010
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$\begin{align}2x+1&> \frac{3x+75}{5}\\5(2x+1)&>3x+75\\10x+5&>3x+75\\10x-3x&>75-5\\7x&>70\\x&>10\\\end{align}$
Question ID: 20230085010
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b) $3-2x> \frac{6+4x}{8}$
a
$x<\frac{9}{10}$
Question ID: 20230085020
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$\begin{align}3-2x&> \frac{6+4x}{8}\\8(3-2x)&>6+4x\\24-16x&>6+4x\\24-6&>4x+16x\\18&>20x\\\frac{18}{20}&>x\\\frac{9}{10}&>x\\x&<\frac{9}{10}\\\end{align}$
Question ID: 20230085020
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c) $4-6x\ge\frac{8x-1}{-5}$
a
$x\le\frac{19}{22}$
Question ID: 20230085030
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$\begin{align}4-6x&\ge\frac{8x-1}{-5}\\-5(4-6x)&\le8x-1\\-20+30x&\le8x-1\\30x-8x&\le-1+20\\22x&\le19\\x&\le\frac{19}{22}\end{align}$
Question ID: 20230085030
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Question 11
Solve the following, showing your solution on a number line
a) $5(x-8)+2(x-1)>0$
a
$x>6$
Question ID: 20230090010
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Question ID: 20230090010
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b) $4(x+5)+6(x+10)\le0$
a
$x\le-8$
Question ID: 20230090020
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c) $7(x-4)-(x+2)\ge0$
a
$x\ge5$
Question ID: 20230090030
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d) $3(2x+1)-4(x-3) < 16$
a
$x < \frac12$
Question ID: 20230090040
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e) $0\le4(4x-1)-3(5x+4)$
a
$16\le x$
or
$x\ge16$
Question ID: 20230090050
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f) $7(2x-3)\ge2(x+5)-4(6x-3)$
a
$x\ge2$
Question ID: 20230090060
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$\begin{align}7(2x-3)-29&\ge2(x+5)-4(6x-3)\\14x-21-29&\ge2x+10-24x+12\\14x-50&\ge22-22x\\36x&\ge72\\x&\ge2\end{align}$
Question ID: 20230090060
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Question 12
Solve the following, showing your solution on a number line
a) $2x+4<3x<2x+9$
a
$4 < x < 9$
Question ID: 20230100010
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Question ID: 20230100010
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b) $7x-2\le8x\le7x+10$
a
$-2\le x\le10$
Question ID: 20230100020
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c) $8x+8<10x\le8x+20$
a
$4 < x\le10$
Question ID: 20230100030
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d) $5x-9\le11x-3<5x+27$
a
$-1\le x<5$
Question ID: 20230100040
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e) $2(3x+5)\le7x\le3(2x+16)$
a
$10\le x\le48$
Question ID: 20230100050
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f) $9(8+4x)<40x+4\le12(3x+7)$
a
$17 < x\le20$
Question ID: 20230100060
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Question 13
By forming an inequation, solve the following
a) A bag contains only red balls or black balls. Out of a bag of $10$ balls. there are more red balls than black balls. How many red balls could there be?
a
There must be more than $5$ red balls in the bag
Question ID: 20230110010
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A bag contains only red balls or black balls. Out of a bag of $10$ balls. there are more red balls than black balls. How many red balls could there be?
Let the number of red balls be $r$ and the number of black balls be $b$
Then
$r>b$
also
$r+b=10$
so
$b=10-r$
Therefore
$\begin{align}r&>10-r\\r+r&>10\\2r&>10\\r&>10\div2\\r&>5\end{align}$
There must be more than $5$ red balls in the bag
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b) A bag contains only green balls or yellow balls. Out of a bag of $50$ balls. there are more green balls than yellow balls. How many green balls could there be?
a
There must be more than $25$ green balls in the bag
Question ID: 20230110020
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A bag contains only green balls or yellow balls. Out of a bag of $50$ balls. there are more green balls than yellow balls. How many green balls could there be?
Let the number of green balls be $g$ and the number of yellow balls be $y$
Then
$g>y$
also
$g+y=50$
so
$y=50-g$
Therefore
$\begin{align}g&>50-g\\g+g&>50\\2g&>50\\g&>50\div2\\r&>25\end{align}$
There must be more than $25$ green balls in the bag
Question ID: 20230110020
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Question 1
Solve the following, showing your solution on a number line
a) $2x>x+3$
b) $6x>5x+7$
c) $5+10x\le11x$
d) $29x+10\ge30x$
e) $12y-4>13y$
f) $-6z<16-7z$
Question 2
Solve the following, showing your solution on a number line
a) $3x\ge x+8$
b) $11x\ge4x-42$
c) $36-2x>4x$
d) $4m<19-15m$
e) $22-x\le 43x$
f) $11a\le -100-9a$
Question 3
Solve the following, showing your solution on a number line
a) $\frac{x+6}{3}\le x$
b) $\frac{30-4x}{2}\ge x$
c) $-12x> \frac{4x+100}{3}$
Question 4
Solve the following, showing your solution on a number line
a) $x+4\le2x+3$
b) $5x+7\ge4x+10$
c) $18x-5<17x+80$
d) $4-6x>9-7x$
e) $6-4a<8-3a$
f) $-12-16h<-17h-44$
Question 5
Solve the following, showing your solution on a number line
a) $15+5x\ge3x+25$
b) $20+7x\le10x+17$
c) $12x-3<19+20x$
d) $2w-25>8-9w$
e) $16-7n\le10-10n$
f) $8x+20>-9x-31$
Question 6
Solve the following, showing your solution on a number line
a) $2(x+4)\ge x+10$
b) $5(x+3)<4x+55$
c) $3(x-4)<2x$
d) $4+10x\ge9(x+3)$
e) $16-4x>5(4-x)$
f) $-8(x+6)\le20-7x$
Question 7
Solve the following, showing your solution on a number line
a) $5(x+1)>2x+14$
b) $6x+44>2(x+30)$
c) $3(x-3)\le11+5x$
d) $3(2x+10)\ge15x+3$
e) $5-7x\le-2(3-2x)$
f) $-5(12-4x)\ge-10x+90$
Question 8
Solve the following, showing your solution on a number line
a) $4(x+6)>5(x+2)$
b) $4(2x+3)\le7(x+5)$
c) $7(3x-3)>11(2x-4)$
d) $6(4+x)\le-5(2-x)$
e) $-5(6-5x)>-8(-3x-5)$
f) $-9(7-9x)\le4(20x-16)$
Question 9
Solve the following, showing your solution on a number line
a) $6(x+4)\ge4(x+9)$
b) $10(x+6)\ge15(x+4)$
c) $8(x-4)<14(x+5)$
d) $4(3x+2)>-3(4+x)$
e) $5(5-4x)<5(2x-1)$
f) $-5(2-x)\ge7(5+5x)$
Question 10
Solve the following, showing your solution on a number line
a) $2x+1> \frac{3x+75}{5}$
b) $3-2x> \frac{6+4x}{8}$
c) $4-6x\ge\frac{8x-1}{-5}$
Question 11
Solve the following, showing your solution on a number line
a) $5(x-8)+2(x-1)>0$
b) $4(x+5)+6(x+10)\le0$
c) $7(x-4)-(x+2)\ge0$
d) $3(2x+1)-4(x-3) < 16$
e) $0\le4(4x-1)-3(5x+4)$
f) $7(2x-3)\ge2(x+5)-4(6x-3)$
Question 12
Solve the following, showing your solution on a number line
a) $2x+4<3x<2x+9$
b) $7x-2\le8x\le7x+10$
c) $8x+8<10x\le8x+20$
d) $5x-9\le11x-3<5x+27$
e) $2(3x+5)\le7x\le3(2x+16)$
f) $9(8+4x)<40x+4\le12(3x+7)$
Question 13
By forming an inequation, solve the following
a) A bag contains only red balls or black balls. Out of a bag of $10$ balls. there are more red balls than black balls. How many red balls could there be?
b) A bag contains only green balls or yellow balls. Out of a bag of $50$ balls. there are more green balls than yellow balls. How many green balls could there be?
Answers
Question 1
a) $x>3$

b) $x>7$

c) $5\le x$
or
$x\ge5$

d) $10\ge x$
or
$x\le10$

e) $-4>y$
or
$y<-4$

f) $z<16$

Question 2
a) $x\ge4$
b) $x\ge-6$
c) $6>x$
or
$x<6$
d) $x<1$
e) $\frac12\le x$
or
$x\ge\frac12$
f) $x\le-5$
Question 3
a) $x\ge3$
b) $x\le5$
c) $x<-\frac52$
Question 4
a) $1\le x$
or
$x\ge1$
b) $x\ge3$
c) $x<85$
d) $x>5$
e) $-2 < a$
or
$a>-2$
f) $h<-32$
Question 5
a) $x\ge5$
b) $1\le x$
or
$x\ge1$
c) $2 < x$
or
$x>2$
d) $w>3$
e) $n\le-2$
f) $x>-3$
Question 6
a) $x\ge2$
b) $x<40$
c) $x<12$
d) $x\ge23$
e) $x>4$
f) $-68\le x$
or
$x\ge-68$
Question 7
a) $x>3$
b) $x>4$
c) $-10\le x$
or
$x\ge-10$
d) $3\ge x$
or
$x\le3$
e) $1\le x$
or
$x\ge1$
f) $x\ge5$
Question 8
a) $14>x$
or
$x<14$
b) $x\le23$
c) $23>x$
or
$x<23$
d) $x\le-34$
e) $x>70$
f) $x\le-1$
Question 9
a) $x\ge6$
b) $0\ge x$
or
$x\le0$
c) $-17>x$
or
$x<-17$
d) $x>-\frac34$
e) $x>1$
f) $-\frac32\ge x$
or
$x\le-\frac32$
Question 10
a) $x>10$
b) $x<\frac{9}{10}$
c) $x\le\frac{19}{22}$
Question 11
a) $x>6$
b) $x\le-8$
c) $x\ge5$
d) $x < \frac12$
e) $16\le x$
or
$x\ge16$
f) $x\ge2$
Question 12
a) $4 < x < 9$

b) $-2\le x\le10$

c) $4 < x\le10$

d) $-1\le x<5$

e) $10\le x\le48$

f) $17 < x\le20$

Question 13
a) There must be more than $5$ red balls in the bag
b) There must be more than $25$ green balls in the bag
Question 1
Solve the following, showing your solution on a number line
a) $2x>x+3$
b) $6x>5x+7$
c) $5+10x\le11x$
d) $29x+10\ge30x$
e) $12y-4>13y$
f) $-6z<16-7z$
Question 2
Solve the following, showing your solution on a number line
a) $3x\ge x+8$
b) $11x\ge4x-42$
c) $36-2x>4x$
d) $4m<19-15m$
e) $22-x\le 43x$
f) $11a\le -100-9a$
Question 3
Solve the following, showing your solution on a number line
a) $\frac{x+6}{3}\le x$
b) $\frac{30-4x}{2}\ge x$
c) $-12x> \frac{4x+100}{3}$
Question 4
Solve the following, showing your solution on a number line
a) $x+4\le2x+3$
b) $5x+7\ge4x+10$
c) $18x-5<17x+80$
d) $4-6x>9-7x$
e) $6-4a<8-3a$
f) $-12-16h<-17h-44$
Question 5
Solve the following, showing your solution on a number line
a) $15+5x\ge3x+25$
b) $20+7x\le10x+17$
c) $12x-3<19+20x$
d) $2w-25>8-9w$
e) $16-7n\le10-10n$
f) $8x+20>-9x-31$
Question 6
Solve the following, showing your solution on a number line
a) $2(x+4)\ge x+10$
b) $5(x+3)<4x+55$
c) $3(x-4)<2x$
d) $4+10x\ge9(x+3)$
e) $16-4x>5(4-x)$
f) $-8(x+6)\le20-7x$
Question 7
Solve the following, showing your solution on a number line
a) $5(x+1)>2x+14$
b) $6x+44>2(x+30)$
c) $3(x-3)\le11+5x$
d) $3(2x+10)\ge15x+3$
e) $5-7x\le-2(3-2x)$
f) $-5(12-4x)\ge-10x+90$
Question 8
Solve the following, showing your solution on a number line
a) $4(x+6)>5(x+2)$
b) $4(2x+3)\le7(x+5)$
c) $7(3x-3)>11(2x-4)$
d) $6(4+x)\le-5(2-x)$
e) $-5(6-5x)>-8(-3x-5)$
f) $-9(7-9x)\le4(20x-16)$
Question 9
Solve the following, showing your solution on a number line
a) $6(x+4)\ge4(x+9)$
b) $10(x+6)\ge15(x+4)$
c) $8(x-4)<14(x+5)$
d) $4(3x+2)>-3(4+x)$
e) $5(5-4x)<5(2x-1)$
f) $-5(2-x)\ge7(5+5x)$
Question 10
Solve the following, showing your solution on a number line
a) $2x+1> \frac{3x+75}{5}$
b) $3-2x> \frac{6+4x}{8}$
c) $4-6x\ge\frac{8x-1}{-5}$
Question 11
Solve the following, showing your solution on a number line
a) $5(x-8)+2(x-1)>0$
b) $4(x+5)+6(x+10)\le0$
c) $7(x-4)-(x+2)\ge0$
d) $3(2x+1)-4(x-3) < 16$
e) $0\le4(4x-1)-3(5x+4)$
f) $7(2x-3)\ge2(x+5)-4(6x-3)$
Question 12
Solve the following, showing your solution on a number line
a) $2x+4<3x<2x+9$
b) $7x-2\le8x\le7x+10$
c) $8x+8<10x\le8x+20$
d) $5x-9\le11x-3<5x+27$
e) $2(3x+5)\le7x\le3(2x+16)$
f) $9(8+4x)<40x+4\le12(3x+7)$
Question 13
By forming an inequation, solve the following
a) A bag contains only red balls or black balls. Out of a bag of $10$ balls. there are more red balls than black balls. How many red balls could there be?
b) A bag contains only green balls or yellow balls. Out of a bag of $50$ balls. there are more green balls than yellow balls. How many green balls could there be?
Answers
Question 1
a) $x>3$

b) $x>7$

c) $5\le x$
or
$x\ge5$

d) $10\ge x$
or
$x\le10$

e) $-4>y$
or
$y<-4$

f) $z<16$

Question 2
a) $x\ge4$
b) $x\ge-6$
c) $6>x$
or
$x<6$
d) $x<1$
e) $\frac12\le x$
or
$x\ge\frac12$
f) $x\le-5$
Question 3
a) $x\ge3$
b) $x\le5$
c) $x<-\frac52$
Question 4
a) $1\le x$
or
$x\ge1$
b) $x\ge3$
c) $x<85$
d) $x>5$
e) $-2 < a$
or
$a>-2$
f) $h<-32$
Question 5
a) $x\ge5$
b) $1\le x$
or
$x\ge1$
c) $2 < x$
or
$x>2$
d) $w>3$
e) $n\le-2$
f) $x>-3$
Question 6
a) $x\ge2$
b) $x<40$
c) $x<12$
d) $x\ge23$
e) $x>4$
f) $-68\le x$
or
$x\ge-68$
Question 7
a) $x>3$
b) $x>4$
c) $-10\le x$
or
$x\ge-10$
d) $3\ge x$
or
$x\le3$
e) $1\le x$
or
$x\ge1$
f) $x\ge5$
Question 8
a) $14>x$
or
$x<14$
b) $x\le23$
c) $23>x$
or
$x<23$
d) $x\le-34$
e) $x>70$
f) $x\le-1$
Question 9
a) $x\ge6$
b) $0\ge x$
or
$x\le0$
c) $-17>x$
or
$x<-17$
d) $x>-\frac34$
e) $x>1$
f) $-\frac32\ge x$
or
$x\le-\frac32$
Question 10
a) $x>10$
b) $x<\frac{9}{10}$
c) $x\le\frac{19}{22}$
Question 11
a) $x>6$
b) $x\le-8$
c) $x\ge5$
d) $x < \frac12$
e) $16\le x$
or
$x\ge16$
f) $x\ge2$
Question 12
a) $4 < x < 9$

b) $-2\le x\le10$

c) $4 < x\le10$

d) $-1\le x<5$

e) $10\le x\le48$

f) $17 < x\le20$

Question 13
a) There must be more than $5$ red balls in the bag
b) There must be more than $25$ green balls in the bag
Solutions
Question 1
a) $\begin{align}2x&>x+3\\2x-x&>3\\x&>3\end{align}$

b)
c) $\begin{align}5+10x&\le11x\\5&\le11x-10x\\5&\le x\\x&\ge5\end{align}$

d)
e)
f)
Question 2
a) $\begin{align}3x&\ge x+8\\3x-x&\ge8\\2x&\ge8\\x&\ge8\div2\\x&\ge4\end{align}$
b) $\begin{align}11x&\ge4x-42\\11x-4x&\ge-42\\7x&\ge-42\\x&\ge-42\div7\\x&\ge-6\end{align}$
c)
d)
e)
f)
Question 3
a) $\begin{align}\frac{x+6}{3}&\le x\\x+2&\le3x\\2&\le3x-x\\6&\le2x\\3&\le x\\x&\ge3\end{align}$
b) $\begin{align}\frac{18-4x}{2}&\ge x\\18-4x&\ge2x\\18&\ge6x\\3&\ge x\\x&\le3\end{align}$
c) $\begin{align}-12x&> \frac{4x+100}{3}\\-36x&>4x+100\\-40x&>100\\x&<\frac{100}{-40}\\x&<-\frac52\\\end{align}$
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) $\begin{align}2x+1&> \frac{3x+75}{5}\\5(2x+1)&>3x+75\\10x+5&>3x+75\\10x-3x&>75-5\\7x&>70\\x&>10\\\end{align}$
b) $\begin{align}3-2x&> \frac{6+4x}{8}\\8(3-2x)&>6+4x\\24-16x&>6+4x\\24-6&>4x+16x\\18&>20x\\\frac{18}{20}&>x\\\frac{9}{10}&>x\\x&<\frac{9}{10}\\\end{align}$
c) $\begin{align}4-6x&\ge\frac{8x-1}{-5}\\-5(4-6x)&\le8x-1\\-20+30x&\le8x-1\\30x-8x&\le-1+20\\22x&\le19\\x&\le\frac{19}{22}\end{align}$
Question 11
a)
b)
c)
d)
e)
f) $\begin{align}7(2x-3)-29&\ge2(x+5)-4(6x-3)\\14x-21-29&\ge2x+10-24x+12\\14x-50&\ge22-22x\\36x&\ge72\\x&\ge2\end{align}$
Question 12
a)
b)
c)
d)
e)
f)
Question 13
a) A bag contains only red balls or black balls. Out of a bag of $10$ balls. there are more red balls than black balls. How many red balls could there be?
Let the number of red balls be $r$ and the number of black balls be $b$
Then
$r>b$
also
$r+b=10$
so
$b=10-r$
Therefore
$\begin{align}r&>10-r\\r+r&>10\\2r&>10\\r&>10\div2\\r&>5\end{align}$
There must be more than $5$ red balls in the bag
b) A bag contains only green balls or yellow balls. Out of a bag of $50$ balls. there are more green balls than yellow balls. How many green balls could there be?
Let the number of green balls be $g$ and the number of yellow balls be $y$
Then
$g>y$
also
$g+y=50$
so
$y=50-g$
Therefore
$\begin{align}g&>50-g\\g+g&>50\\2g&>50\\g&>50\div2\\r&>25\end{align}$
There must be more than $25$ green balls in the bag