Simultaneous inequalities questions with answers

Questions

1. Question 1a < px + q ≤ b

   Solve $11 < 2x + 5 \leq 19$.

   Show the worked solution

   1. Step 1: Do the same to all three parts, so the x term is alone in the middle$6 < 2x \leq 14$
   2. Step 2: Divide all three parts by the number in front of x. Do the signs change?$3 < x \leq 7$
2. Question 2Negative x term in the middle

   Solve $21 < -3x + 9 < 27$.

   Show the worked solution

   1. Step 1: Do the same to all three parts, so the x term is alone in the middle$12 < -3x < 18$
   2. Step 2: Divide all three parts by the number in front of x, keeping each number in its place. Do the signs change?$-4 > x > -6$ (dividing by a negative number reverses both signs)
   3. Step 3: Turn it round so the smaller number is on the left (each sign turns round with it)$-6 < x < -4$
3. Question 3Two inequalities with “and”

   Solve $-4x + 4 < 28$ and $-x + 1 \geq 4$, or write "no solution" if there is none.

   Show the worked solution

   1. Step 1: Solve the first inequality$-4x + 4 < 28$ gives $x > -6$
   2. Step 2: Solve the second inequality$-x + 1 \geq 4$ gives $x \leq -3$
   3. Step 3: Do the two ranges overlap?Yes: they overlap.
   4. Step 4: Write the answer$-6 < x \leq -3$
4. Question 4…or no solution

   Solve $-3x - 6 \geq -30$ and $3x + 3 \geq 18$, or write "no solution" if there is none.

   Show the worked solution

   1. Step 1: Solve the first inequality$-3x - 6 \geq -30$ gives $x \leq 8$
   2. Step 2: Solve the second inequality$3x + 3 \geq 18$ gives $x \geq 5$
   3. Step 3: Do the two ranges overlap?Yes: they overlap.
   4. Step 4: Write the answer$5 \leq x \leq 8$
5. Question 5A chain with fractions: split it into two

   Solve the inequality $\dfrac{2x - 1}{4} < \dfrac{x + 4}{5} \leq x + 8$.

   Show the worked solution

   1. Step 1: Split the chain into two inequalities: the middle part goes in both$\dfrac{2x - 1}{4} < \dfrac{x + 4}{5}$ and $\dfrac{x + 4}{5} \leq x + 8$
   2. Step 2: The first inequality: multiply both sides by the LCM of its denominators, collect the x terms, then solve$\dfrac{2x - 1}{4} < \dfrac{x + 4}{5}$Multiply by $20$: $5(2x - 1) < 4(x + 4)$$6x < 21$$x < \dfrac{7}{2}$
   3. Step 3: The second inequality: multiply both sides by the LCM of its denominators, collect the x terms, then solve$\dfrac{x + 4}{5} \leq x + 8$Multiply by $5$: $x + 4 \leq 5(x + 8)$$-4x \leq 36$$x \geq -9$ (dividing by a negative number reverses the sign)
   4. Step 4: Both must be true: write the overlap as one range$-9 \leq x < \dfrac{7}{2}$