Sec 2 Maths · Linear equations, inequalities and simultaneous equations

Integer solutions questions with answers

Practice questions on integer solutions, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Write down all the integer values of $x$ such that $-4 \leq x < 1$ .

Step 1 of 2

Check each end: is it included?

-4 \leq x < 1

: include the left end, leave out the right end.

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Questions

Question 1Integers in a range
Write down all the integer values of $x$ such that $-\dfrac{5}{2} < x \leq 0$ .
Show the worked solution
1. Step 1: Check each end: is it included? $-\dfrac{5}{2} < x \leq 0$ : leave out the left end, include the right end.
2. Step 2: List the integers $-2$ , $-1$ , $0$
Question 2Greatest or smallest integer
Find the largest integer value of $x$ that satisfies $5x + 9 \leq -11$ .
Show the worked solution
1. Step 1: Collect the x terms on one side and the numbers on the other $5x \leq -20$
2. Step 2: Divide both sides by the number in front of x. Does the sign change? $x \leq -4$
3. Step 3: Find the largest integerIn $x \leq -4$ , the largest integer is $-4$ .
Question 3…with the sign reversed
Find the largest integer value of $x$ that satisfies $3 - 3x > 9$ .
Show the worked solution
1. Step 1: Collect the x terms on one side and the numbers on the other $-3x > 6$
2. Step 2: Divide both sides by the number in front of x. Does the sign change? $x < -2$ (dividing by a negative number reverses the sign)
3. Step 3: Find the largest integerIn $x < -2$ , the largest integer is $-3$ .
Question 4The smallest prime, the largest perfect square…
Solve the inequality $\dfrac{2x + 5}{3} > \dfrac{3x + 3}{4}$ . Hence write down the largest prime number that satisfies the inequality.
Show the worked solution
1. Step 1: Multiply both sides by the LCM of the denominators $4(2x + 5) > 3(3x + 3)$
2. Step 2: Expand the brackets $8x + 20 > 9x + 9$
3. Step 3: Collect the x terms on one side and the numbers on the other $-x > -11$
4. Step 4: Divide both sides by the number in front of x. Does the sign change? $x < 11$ (dividing by a negative number reverses the sign)
5. Step 5: Go down from the end of the range until you reach a prime numberIn $x < 11$ : $10$ , $9$ , $8$ , $7$ , and only $7$ is a prime number.

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Questions

Question 1Integers in a range

Write down all the integer values of $x$ such that $-\dfrac{5}{2} < x \leq 0$ .

Show the worked solution

Step 1: Check each end: is it included? $-\dfrac{5}{2} < x \leq 0$ : leave out the left end, include the right end.
Step 2: List the integers $-2$ , $-1$ , $0$

Question 2Greatest or smallest integer

Find the largest integer value of $x$ that satisfies $5x + 9 \leq -11$ .

Show the worked solution

Step 1: Collect the x terms on one side and the numbers on the other $5x \leq -20$
Step 2: Divide both sides by the number in front of x. Does the sign change? $x \leq -4$
Step 3: Find the largest integerIn $x \leq -4$ , the largest integer is $-4$ .

Question 3…with the sign reversed

Find the largest integer value of $x$ that satisfies $3 - 3x > 9$ .

Show the worked solution

Step 1: Collect the x terms on one side and the numbers on the other $-3x > 6$
Step 2: Divide both sides by the number in front of x. Does the sign change? $x < -2$ (dividing by a negative number reverses the sign)
Step 3: Find the largest integerIn $x < -2$ , the largest integer is $-3$ .

Question 4The smallest prime, the largest perfect square…

Solve the inequality $\dfrac{2x + 5}{3} > \dfrac{3x + 3}{4}$ . Hence write down the largest prime number that satisfies the inequality.

Show the worked solution

Step 1: Multiply both sides by the LCM of the denominators $4(2x + 5) > 3(3x + 3)$
Step 2: Expand the brackets $8x + 20 > 9x + 9$
Step 3: Collect the x terms on one side and the numbers on the other $-x > -11$
Step 4: Divide both sides by the number in front of x. Does the sign change? $x < 11$ (dividing by a negative number reverses the sign)
Step 5: Go down from the end of the range until you reach a prime numberIn $x < 11$ : $10$ , $9$ , $8$ , $7$ , and only $7$ is a prime number.