Integer solutions questions with answers

Questions

1. Question 1Integers in a range

   Write down all the integer values of $x$ such that $2 < x < 4$.

   Show the worked solution

   1. Step 1: Check each end: is it included?$2 < x < 4$: leave out the left end, leave out the right end.
   2. Step 2: List the integers$3$
2. Question 2Greatest or smallest integer

   Find the smallest integer value of $x$ that satisfies $2x + 5 > -3$.

   Show the worked solution

   1. Step 1: Collect the x terms on one side and the numbers on the other$2x > -8$
   2. Step 2: Divide both sides by the number in front of x. Does the sign change?$x > -4$
   3. Step 3: Find the smallest integerIn $x > -4$, the smallest integer is $-3$.
3. Question 3…with the sign reversed

   Find the smallest integer value of $x$ that satisfies $11 - 3x < -4$.

   Show the worked solution

   1. Step 1: Collect the x terms on one side and the numbers on the other$-3x < -15$
   2. Step 2: Divide both sides by the number in front of x. Does the sign change?$x > 5$ (dividing by a negative number reverses the sign)
   3. Step 3: Find the smallest integerIn $x > 5$, the smallest integer is $6$.
4. Question 4Integers satisfying two inequalities

   Write down all the integer values of $x$ that satisfy $4x - 1 \leq 33$ and $-3x + 6 < -6$.

   Show the worked solution

   1. Step 1: Solve the first inequality$4x - 1 \leq 33$ gives $x \leq \dfrac{17}{2}$
   2. Step 2: Solve the second inequality$-3x + 6 < -6$ gives $x > 4$
   3. Step 3: Do the two ranges overlap?Yes: they overlap.
   4. Step 4: Write the answer$4 < x \leq \dfrac{17}{2}$
   5. Step 5: List the integers in the range$5$, $6$, $7$, $8$
5. Question 5The smallest prime, the largest perfect square…

   Solve the inequality $3x - 1 < x + 9$. Hence write down the largest prime number that satisfies the inequality.

   Show the worked solution

   1. Step 1: Collect the x terms on one side and the numbers on the other$2x < 10$
   2. Step 2: Divide both sides by the number in front of x. Does the sign change?$x < 5$
   3. Step 3: Go down from the end of the range until you reach a prime numberIn $x < 5$: $4$, $3$, and only $3$ is a prime number.
6. Question 6Two integers in ranges: the smallest or largest value of y² − x²

   $x$ and $y$ are integers such that $1 \leq x < 5$ and $-1 \leq y \leq 4$. Find the smallest possible value of $y^2 - x^2$.

   Show the worked solution

   1. Step 1: List the integers each letter can be$x$: $1$, $2$, $3$, $4$$y$: $-1$, $0$, $1$, $2$, $3$, $4$
   2. Step 2: Make y² as small as possible: square each y (a negative number squared is positive)$y^2 = 0$
   3. Step 3: Make x² as large as possible$x^2 = 16$
   4. Step 4: Put them together$0 - 16 = -16$