Algebra - Solving Inequalities with Unknowns on Both Sides

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

---