Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Lesson 5 of this unit. Is 4 a solution of x > 4? Of x ≥ 4?
Answer
No for x > 4, because 4 is not greater than 4. Yes for x ≥ 4. - From Unit 3 (integers), lesson 4. Work out 28 ÷ (−4).
Answer
−7. A positive divided by a negative is negative. - From Lesson 3 of this unit. Solve −5x + 3 = 18.
Answer
x = −3. Subtract 3: −5x = 15, then divide by −5.
Words used in this lesson
- reverse
- turn the other way round. For example: < becomes >.
- negative number
- a number less than zero. For example: −4.
- coefficient
- the number multiplying a letter. For example: −3 in −3x.
- test value
- a number you try to see if it works. For example: x = 0.
- boundary
- the number where the range starts. For example: −7 in x > −7.
Bigger, but worse
Owing 8 thousand naira is worse than owing 2 thousand, even though 8 > 2. Write the debts as negative numbers and compare −8 and −2.
Predict first: start from 6 > 2. Does adding a negative number, such as −4, change which is bigger? What about multiplying both by a negative number? Test each with the buttons.
6 > 2. Choose an operation to apply to both numbers.
Show me the steps
- 6 > 2, add −4 to both: 2 > −2. The order stays when you add.
- 6 > 2, multiply both by 3: 18 > 6. The order stays when you multiply by a positive number.
- 6 > 2, multiply both by −3: −18 < −6. The order reverses.
- 6 > 2, divide both by −2: −3 < −1. Dividing by a negative number reverses it too.
- 6 > 2, multiply both by 0: both become 0, so nothing about the order is left.
Multiplying or dividing both sides by a negative number reverses the sign. < becomes >, ≤ becomes ≥, and the other way round.
Adding or subtracting any number, or multiplying or dividing by a positive number, keeps the sign.
Why: multiplying by a negative number flips the number line over zero (and stretches or shrinks it), so the number that was further right ends up further left.
Test the answer
A value taken from your answer must make the original inequality true, and a value outside it must make it false. Test a value against the reversed answer and against the answer you get if you forget to reverse.
−3x < 21 with t = 0: the left side is 0, and 0 < 21 is true. The reversed answer x > −7 includes 0; the unreversed x < −7 does not include 0.
Show me the steps
- −3x < 21, test t = 0: 0 < 21 is true. The reversed answer x > −7 includes 0; the unreversed x < −7 does not.
- Test t = −9: −3 × (−9) = 27, and 27 < 21 is false. The reversed answer x > −7 leaves out −9, so it agrees; the unreversed one includes −9, so it does not.
- Test t = −7, the boundary: −3 × (−7) = 21, and 21 < 21 is false. Both rays leave the boundary out: the circle is open.
- Pick inequality 3, 5 − x > 1: the answer is x < 4. Test t = 5: 5 − 5 = 0, and 0 > 1 is false, which agrees with 5 being outside x < 4.
Reverse only for a negative multiplier or divisor. In x − 3 > −5 you add 3 to both sides and get x > −2: the numbers are negative but nothing was multiplied or divided, so the sign stays. And in −3x > 12 the negative is attached to the x, so it is the divisor −3 that reverses the sign.
Worked examples
Solve −5x < 40.
- Divide both sides by −5 and reverse the sign: x > −8.
- Check: x = 0 gives 0 < 40, true. x = −9 gives 45 < 40, false. Both agree with x > −8.
Solve 3 − 4x ≥ 15.
- Subtract 3 from both sides: −4x ≥ 12.
- Divide both sides by −4 and reverse the sign: x ≤ −3.
- Check: x = −5 gives 3 + 20 = 23 ≥ 15, true.
Solve 5 − x > 2 without dividing by a negative number.
- Add x to both sides: 5 > 2 + x.
- Subtract 2 from both sides: 3 > x, which says x < 3. Reading 3 > x as x < 3 only swaps the two sides; nothing was multiplied by a negative number.
Your go. The first step is given. Solve −7x ≤ 42.
- Divide both sides by −7 and reverse the sign: x ≥ ?
- Find the number, then test one value from your answer.
Check the answer
42 ÷ (−7) = −6, so x ≥ −6. Test x = 0: 0 ≤ 42, true.
Check before you go on
1. Start with 8 > 5. Multiply both sides by −3. Which is true?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Solve −4x < 20, giving x > k. What is k?
3. Solve −8x ≥ 16. Which is the solution?
Level 2 · Building
4. Solve 6 − 2x ≤ 14, giving x ≥ k. What is k?
5. Solve 10 − 3x > 1. What is the largest whole number x can be?
6. Which inequality is equivalent to −x ≤ 7?
Level 3 · Stretch
7. Solve 2(3 − x) > 10, giving x < k. What is k?
8. A drone is 40 m above the ground and descends 4 m every second, so its height after t seconds is 40 − 4t metres. What is the largest whole number of seconds t for which it is still at least 12 m above the ground?
Something to think about
Adding −4 to both sides of 6 > 2 keeps the order, but multiplying by −4 reverses it. Use the first figure to explain, in your own words, why the two are different.
One way to think about it
Adding the same number moves both points the same distance in the same direction, so the gap between them and their order do not change. Multiplying by a negative number reflects the number line through zero: the point that was further to the right ends up further to the left, so the order reverses.
Exit ticket
9. Mei solves −2x < 6 and writes x < −3. Which reply is right?