LemonmathSelf-study sheet

Grade 9 · Lesson 2.3

Linear inequalities in one unknown

Name: Date: Time spent: min

Goals

Recognise a linear inequality in one unknown and check whether a number is a solution.
Solve linear inequalities in one unknown by moving terms and by multiplying both sides.
Reduce inequalities to linear form and solve real-life problems.

Remember

Linear inequality in one unknown: $ax+b<0$ (or $>$ , $\le$ , $\ge$ ), $a\neq0$ .
Moving a term to the other side changes its sign.
Multiplying or dividing by a positive number: keep the direction; by a negative number: reverse it.
Brackets or fractions: expand, multiply by a positive common denominator, reach $ax<c$ , then divide by $a$ .
Real problems: check the conditions (whole numbers…) before choosing the answer.

Exercises

Exercise 1

Which of the numbers $-1$ , $0$ , $3$ are solutions of the inequality $2x+1>0$ ?

Exercise 2

Solve the inequality $4x-8<0$ .

Exercise 3

Solve the inequality $-3x+9\le0$ .

Exercise 4

Solve the inequality $2(x+3)>5x-6$ .

Exercise 5

Solve the inequality $\dfrac{2x-1}{3}\ge\dfrac{x+2}{4}$ .

Exercise 6

A taxi firm charges a starting fee of $10$ thousand đồng plus $14$ thousand đồng per kilometre. With $150$ thousand đồng, what is the greatest distance you can travel (in whole kilometres)?

Exercise 7

Nam has scored $7$ , $8$ and $6$ in three tests. What is the lowest mark he needs in the fourth test so that his mean for the four tests is not below $7.5$ ?

Exercise 8 Review

Find the solutions of the equation $(x-4)(2x+1)=0$ that satisfy the inequality $3x-2>0$ .

Exercise 9

Consider the expressions $A=3(x-1)$ and $B=x+5$ .

a) Find $x$ such that $A=B$ .

b) Find $x$ such that $A>B$ .

c) Find the least integer $x$ such that $A>B$ .

Exercise 10 Challenge

Phone plan A charges $3$ thousand đồng per minute of calls. Plan B charges $45$ thousand đồng a month plus $1$ thousand đồng per minute. In a month, what is the least number of whole minutes of calls for which plan B is cheaper than plan A?

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. Which is a linear inequality in one unknown?

0x+3>0

.B.

x^2-1<0

.C.

2x-5\le0

.D.

\dfrac1x+2>0

Question 2. The solution of the inequality

-2x+6>0

x>3

.B.

x<3

.C.

x>-3

.D.

x<-3

Question 3. The solution of the inequality

3(x-2)\le x+4

x\le5

.B.

x\ge5

.C.

x\le3

.D.

x\le-1

Question 4. The greatest integer satisfying

5x-12<0

2

.B.

3

.C.

1

.D.

-2

Part II. True or false

Question 1. Consider the inequality

4-2x\ge x-5

a) $x=3$ is a solution.	T / F
b) Moving terms gives $-3x\ge-9$ .	T / F
c) The solution of the inequality is $x\ge3$ .	T / F
d) Exactly $4$ non-negative integers satisfy the inequality.	T / F

Part III. Short answer

Question 1. Find the least integer satisfying

2(x+1)>9-x

Question 2. A taxi firm charges a starting fee of

12

thousand đồng plus

15

thousand đồng per kilometre. With

200

thousand đồng, what is the greatest whole number of kilometres you can travel?

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $x_0$ is a solution if substituting it gives a true statement.

Steps:

Substitute each number into $2x+1$ .
Compare with $0$ .

Exercise 2.

What you need: move a term, then divide by a positive number (keep the direction).

Steps:

Move $-8$ to the right-hand side.
Divide by $4$ .

Exercise 3.

What you need: dividing by a negative number reverses the sign.

Steps:

Move $9$ to the right-hand side.
Divide both sides by $-3$ and reverse the sign.

Exercise 4.

What you need: expand, move terms, and watch the sign of the coefficient of $x$ .

Steps:

Expand: $2x+6>5x-6$ .
Moving terms gives $-3x>-12$ .
Divide by $-3$ and reverse the sign.

Exercise 5.

What you need: multiply both sides by the positive common denominator $12$ ; the direction stays the same.

Steps:

Multiply both sides by $12$ .
Expand, move terms, divide by the coefficient of $x$ .

Exercise 6.

What you need: the fare must not exceed the money you have.

Steps:

Let $x$ be the number of kilometres; form $10+14x\le150$ .
Solve and choose the greatest whole number.

Exercise 7.

What you need: "not below" is $\ge$ ; mean $=$ total $\div$ number of tests.

Steps:

Let $x$ be the fourth mark ( $0\le x\le10$ ).
Form $\dfrac{7+8+6+x}{4}\ge7.5$ and multiply both sides by $4$ .

Exercise 8.

What you need: equations in factorised form (lesson 4) and checking solutions of an inequality.

Steps:

Solve the equation in factorised form.
Substitute each solution into $3x-2$ .

Exercise 9.

What you need: solving linear equations and linear inequalities in one unknown.

Steps:

a) Solve $3x-3=x+5$ .
b) Solve $3x-3>x+5$ .
c) Choose the least integer that satisfies part b).

Exercise 10.

What you need: "B is cheaper than A" means the cost of plan B $<$ the cost of plan A.

Steps:

Let $x$ be the number of minutes; plan A costs $3x$ and plan B costs $45+x$ .
Solve $45+x<3x$ .
Choose the least whole number.

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