Topics › KS3 › Algebra

Inequalities

ineq-01 Present from here

$x$ is an integer and $-2 < x \le 3$. Which list shows all the possible values of $x$?

Option A: $-2, -1, 0, 1, 2, 3$

Option B: $-1, 0, 1, 2, 3$

Option C: $-1, 0, 1, 2$

Option D: $0, 1, 2, 3$

Answer and misconceptions

Option A: $-2, -1, 0, 1, 2, 3$ ✗
Includes the end value of a strict inequalityIncludes $-2$, but $<$ means "less than", so $x$ cannot equal $-2$.

Option B: $-1, 0, 1, 2, 3$ ✓ Correct

Option C: $-1, 0, 1, 2$ ✗
Leaves out the end value of $\le$Leaves out $3$, but $\le$ means "less than or equal to", so $x$ can be $3$.

Option D: $0, 1, 2, 3$ ✗
Leaves out the negative integersStarts at $0$, forgetting that $-1$ is an integer greater than $-2$.
ineq-02 Present from here

Solve $3x + 4 > 19$

Option A: $x = 5$

Option B: $x < 5$

Option C: $x > \frac{23}{3}$

Option D: $x > 5$

Answer and misconceptions

Option A: $x = 5$ ✗
Treats the inequality as an equationSolves $3x + 4 = 19$. The answer to an inequality is a range of values, not one value.

Option B: $x < 5$ ✗
Reverses the sign when dividing by a positive numberFlips $>$ to $<$ when dividing by $3$; the sign only reverses when multiplying or dividing by a negative.

Option C: $x > \frac{23}{3}$ ✗
Adds instead of subtracting to undo $+4$Works out $19 + 4 = 23$, then divides by $3$. To undo $+4$, subtract $4$ from both sides.

Option D: $x > 5$ ✓ Correct
ineq-03 Present from here

Solve $-2x < 8$

Option A: $x > -4$

Option B: $x < -4$

Option C: $x < 4$

Option D: $x > -16$

Answer and misconceptions

Option A: $x > -4$ ✓ Correct

Option B: $x < -4$ ✗
Keeps the sign when dividing by a negativeDivides by $-2$ but does not reverse $<$. Check: $x = -5$ gives $-2 \times -5 = 10$, which is not less than $8$.

Option C: $x < 4$ ✗
Ignores the negative coefficientTreats $-2x$ as $2x$ and divides $8$ by $2$.

Option D: $x > -16$ ✗
Multiplies by the coefficient instead of dividingWorks out $8 \times (-2) = -16$ and reverses the sign; to undo $\times(-2)$, divide by $-2$.
ineq-04 Present from here

Which inequality means "$x$ is at least $6$"?

Option A: $x > 6$

Option B: $x \le 6$

Option C: $x \ge 6$

Option D: $x = 6$

Answer and misconceptions

Option A: $x > 6$ ✗
Thinks "at least" leaves out the number itself"At least 6" includes 6, so the sign is $\ge$, not $>$.

Option B: $x \le 6$ ✗
Thinks "at least" means "no more than""At least 6" means 6 or more; $x \le 6$ means 6 or less.

Option C: $x \ge 6$ ✓ Correct

Option D: $x = 6$ ✗
Thinks "at least 6" means exactly 6$x$ can be 6 or any number bigger than 6.
ineq-05 Present from here

$n$ is an integer and $2n + 1 \le 9$. What is the largest possible value of $n$?

Option A: $5$

Option B: $4$

Option C: $3$

Option D: $8$

Answer and misconceptions

Option A: $5$ ✗
Adds instead of subtracting to undo $+1$Gets $2n \le 10$; subtracting 1 from both sides gives $2n \le 8$.

Option B: $4$ ✓ Correct

Option C: $3$ ✗
Thinks $\le$ leaves out the end valueSolves $n \le 4$ but thinks $n$ must be less than 4; $\le$ allows $n = 4$.

Option D: $8$ ✗
Forgets to divide by the coefficientStops at $2n \le 8$ and gives 8; that is the largest value of $2n$, not $n$.