One-variable linear inequalities

(a) $x < 7$

1. Add 5: $3x < 21$. Divide by 3: $x < 7$.

• M1 Writing $3x < 21$.
• A1 The correct answer, $x < 7$.

(b) $-1, 0, 1, 2, 3$

1. $-2$ is not included ($<$), and $3$ is included ($\le$). So $n$ can be $-1, 0, 1, 2, 3$.

• B1 $-1, 0, 1, 2, 3$ and no others.

(a) $x \le 6$

1. Divide by 4: $x - 1 \le 5$. Add 1: $x \le 6$.

• M1 Writing $x - 1 \le 5$ or $4x - 4 \le 20$.
• A1 The correct answer, $x \le 6$.

(b) $6$

1. $x \le 6$ includes 6.

• B1 The correct answer, $6$.

(a) $x<4$

1. Subtract 7 and then divide by minus 3, reversing the inequality.
   $$-3x>-12$$
2. Therefore $x<4$.

• M1 Subtract 7 and then divide by minus 3, reversing the inequality.
• A1 Correct answer: $x<4$

(b) $3$

1. The endpoint is excluded, so move one integer below it.
   $$4-1$$
2. Therefore $3$.

• M1 The endpoint is excluded, so move one integer below it.
• A1 Correct answer: $3$

(a) $n\le 45/8$

1. The fixed charge plus the variable charge cannot exceed the budget.
   $$45+8n\le 90$$
2. Subtract the fixed fee and divide by 8.
   $$8n\le 45$$
3. Therefore $n\le 45/8$.

• P1 The fixed charge plus the variable charge cannot exceed the budget.
• P1 Subtract the fixed fee and divide by 8.
• A1 Correct answer: $n\le 45/8$

(b) $5$

1. Round the upper bound down because guests are whole people.
   $$45/8$$
2. Therefore $5$.

• P1 Round the upper bound down because guests are whole people.
• A1 Correct answer: $5$

(a) −2, −1, 0, 1, 2, 3, 4, 5, 6

1. Subtract 1 from all three expressions and divide by 2.
   $$-5/2<x\le 6$$
2. Therefore −2, −1, 0, 1, 2, 3, 4, 5, 6.

• M1 Subtract 1 from all three expressions and divide by 2.
• A1 Correct answer: −2, −1, 0, 1, 2, 3, 4, 5, 6

(b) $18$

1. Add the finite arithmetic list using its average and count.
   $$9\times (-2+6)/2$$
2. Therefore $18$.

• M1 Add the finite arithmetic list using its average and count.
• A1 Correct answer: $18$

(a) $x \ge -6$

1. Collect the x terms on one side and the numbers on the other.
   $$2x\ge -12$$
2. Therefore $x \ge -6$.

• M1 Collect the x terms on one side and the numbers on the other.
• A1 Correct answer: $x \ge -6$

(b) $-6, -5$

1. Solve the second inequality.
   $$2x<-8$$
2. Therefore $-6, -5$.

• M1 Solve the second inequality.
• A1 Correct answer: $-6, -5$

(a) $x < -3$

1. Subtract 5 from both sides.
   $$-2x > 6$$
2. Divide by $-2$. Dividing by a negative number reverses the inequality sign.
   $$x < -3$$

• M1 Writing $-2x > 6$ (or $5 - 11 > 2x$).
• A1 The correct answer, $x < -3$.

(b) $-4$

1. $x < -3$, so $-3$ itself is not allowed. The largest integer below $-3$ is $-4$.

• B1 The correct answer, $-4$.

(c) $x \le 5$

1. Multiply both sides by 4: $x + 3 \le 8$. Subtract 3: $x \le 5$.

• M1 Writing $x + 3 \le 8$.
• A1 The correct answer, $x \le 5$.

(a) $-3 \le x < 3$

1. Subtract 2 from all three parts.
   $$-9 \le 3x < 9$$
2. Divide all three parts by 3.
   $$-3 \le x < 3$$

• M1 Subtracting 2 from every part: $-9 \le 3x < 9$.
• A1 The correct answer, $-3 \le x < 3$.

(b) $6$

1. First inequality:
   $$4x - 1 > 2x + 6 \Rightarrow 2x > 7 \Rightarrow x > 3.5$$
2. Second inequality:
   $$\frac{x}{2} < 5 \Rightarrow x < 10$$
3. So $3.5 < x < 10$. The integers are $4, 5, 6, 7, 8, 9$: six values.

• P1 Solving the first inequality: $x > 3.5$.
• P1 Solving the second inequality: $x < 10$.
• P1 Combining to $3.5 < x < 10$ and listing the integers.
• A1 The correct answer, $6$.