NCEA Level 1Linear equations and inequationsAS 91947

Equations and inequations: 7 exam questions solved

What this lesson covers

  • Solving is undoing, with the same operation on both sides, so the balance is kept.
  • An inequation has a range of solutions, and multiplying or dividing by a negative reverses its sign.
Key idea

7 questions, one method

  • 1. Expand brackets, clear fractions.
  • 2. Same step on both sides.
  • 3. × or ÷ by a negative: flip the sign.
  • 4. Check in the original.
Your turn

Question 1 · Achieved

  • Solve 4(x − 3) = 2x + 6
Worked example

Q1: 4(x − 3) = 2x + 6

  • Expand first.
  • Check in the original.
  1. 4x − 12 = 2x + 6expand
  2. 2x − 12 = 6−2x both sides
  3. 2x = 18+12 both sides
  4. x = 9÷ 2 both sides
  5. left: 4(9 − 3) = 4 × 6 = 24
  6. right: 2 × 9 + 6 = 24
  7. x = 9
Your turn

Question 2 · Achieved

  • Solve 5 − 2x > 11
Worked example

Q2: 5 − 2x > 11

  • ÷ by −2: flip the sign.
  • Check the end point.
  1. −2x > 6−5 both sides
  2. 3 > 1, but −3 < −1negatives flip order
  3. x < −3÷ −2, > becomes <
  4. x = −4: 5 + 8 = 13 > 11
  5. x = −3: 5 + 6 = 1111 > 11 is false
  6. x < −3
Your turn

Question 3 · Merit

  • Solve (x + 2)/3 − (x − 4)/5 = 2
  • Watch the minus before the 2nd fraction.
Worked example

Q3: two fractions

  • × every term by 15.
  • Numerators in brackets.
  1. LCD of 3 and 5 is 15
  2. 5(x + 2) − 3(x − 4) = 30× 15, every term
  3. 5x + 10 − 3x + 12 = 30−3 × −4 = +12
  4. 2x + 22 = 30collect like terms
  5. 2x = 8−22 both sides
  6. x = 4÷ 2 both sides
  7. 6/3 − 0/5 = 2 − 0 = 2
  8. x = 4
Your turn

Question 4 · Merit

  • Solve 3(2 − x) ≤ x + 14
  • Show the answer on a number line.
Worked example

Q4: 3(2 − x) ≤ x + 14

  • ÷ by −4: ≤ becomes ≥.
  • Is the end included?
  1. 6 − 3x ≤ x + 14expand
  2. 6 − 4x ≤ 14−x both sides
  3. −4x ≤ 8−6 both sides
  4. x ≥ −2÷ −4, ≤ becomes ≥
  5. x = 0: 6 ≤ 14
  6. x = −2: 12 ≤ 12−2 is included
  7. closed dot at −2, shade right
Your turn

Question 5 · Merit, in context

  • Kayak hire in the Bay of Islands.
  • A: $25 plus $8 per hour.
  • B: $40 plus $5 per hour.
  • How many hours for the same cost?
Worked example

Q5: when do they cost the same?

  • Define h, with units.
  • Same cost: set equal.
  1. Let h = number of hours
  2. A: 25 + 8h, B: 40 + 5hcost in dollars
  3. 25 + 8h = 40 + 5hsame cost
  4. 25 + 3h = 40−5h both sides
  5. 3h = 15−25 both sides
  6. h = 5÷ 3 both sides
  7. A: 25 + 8 × 5 = $65
  8. B: 40 + 5 × 5 = $65
  9. 5 hours, both cost $65
Your turn

Question 6 · Excellence

  • Solve −7 < 3 − 2x ≤ 9
  • List the integers that satisfy it.
Worked example

Q6: a double inequation

  • Same step to all three.
  • ÷ by −2: both signs flip.
  1. −10 < −2x ≤ 6−3 from all three
  2. 5 > x ≥ −3÷ −2, both signs flip
  3. −3 ≤ x < 5smallest first
  4. −3 included, 5 not
  5. −3, −2, −1, 0, 1, 2, 3, 4
  6. x = −3: 3 + 6 = 99 ≤ 9, allowed
  7. x = 5: 3 − 10 = −7−7 < −7 is false
Your turn

Question 7 · Excellence, in context

  • A class trip to Auckland Zoo.
  • Bus: $260. Entry: $14 per student.
  • Budget: $900.
  • Greatest number of students?
Worked example

Q7: the zoo trip budget

  • At most: use ≤.
  • Round the way it needs.
  1. Let n = number of students
  2. 260 + 14n ≤ 900cost at most $900
  3. 14n ≤ 640−260 both sides
  4. n ≤ 640 ÷ 14÷ 14, positive: no flip
  5. = 45.71 (2 dp)
  6. round down to 45no part students
  7. n = 45: 260 + 630 = $890
  8. n = 46: 260 + 644 = $904over $900
  9. At most 45 students can go.
Summary

The method that solved all seven

  • Balance both sides, flip for a negative, read the context.
StepHow
Bracketsexpand first, watch the minus
Fractions× every term by the LCD
Balancesame step on both sides
Negatives× or ÷ by one: flip the sign
Wordsdefine the letter and units
Contextround as the question needs

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