Chapter 3 of 8

All 8 chapters
  1. 1Harder linear equations, readunderstood
  2. 2Changing the subject, readunderstood
  3. 3Inequalities, readunderstood
  4. 4Equations of straight lines, readunderstood
  5. 5Two equations: adding or taking away, readunderstood
  6. 6Two equations: substitution, readunderstood
  7. 7Two equations on a graph, readunderstood
  8. 8Word problems and regions, readunderstood

M4.8.3 Chapter 3

Inequalities

The answer to 3x − 4 < 11 is not one number. How do I show every answer?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Solve 3x − 4 < 11 and show it on a number line
  • Solve an inequality where you must flip the sign
  • List the whole numbers that fit −2 < x ≤ 3

A bridge sign says: loads must be less than 5 tonnes. A load of 4 tonnes is allowed. So is 3, and so is 0.5. Many numbers fit. We write the rule as x < 5.

The sign < means "is less than". The sign > means "is more than". The sign ≤ means "is less than or equal to". The sign ≥ means "is more than or equal to". A statement with one of these signs is an inequality.

Draw it first

Show x < 5 on a number line

  1. A number line from −1 to 8. Step 1 puts an open circle at 5. Step 2 draws a thick bar from the circle to the left, ending in an arrow. Step 3 puts dots on the whole numbers −1, 0, 1, 2, 3 and 4. The label x is at the right end.−1012345678x

    1 Draw a number line. Put an open circle at 5. It is open because 5 is not allowed: x is less than 5, not equal to 5.

  2. A number line from −1 to 8. Step 1 puts an open circle at 5. Step 2 draws a thick bar from the circle to the left, ending in an arrow. Step 3 puts dots on the whole numbers −1, 0, 1, 2, 3 and 4. The label x is at the right end.−1012345678x

    2 Draw a thick bar from the circle towards the smaller numbers. The arrow says it goes on for ever.

  3. A number line from −1 to 8. Step 1 puts an open circle at 5. Step 2 draws a thick bar from the circle to the left, ending in an arrow. Step 3 puts dots on the whole numbers −1, 0, 1, 2, 3 and 4. The label x is at the right end.−1012345678x

    3 Dots on the whole numbers 4, 3, 2, 1, 0 and −1 show a few of the numbers that fit. The arrow shows there are more.

An open circle means the end number is not included. It goes with < and >. A filled circle means the end number is included. It goes with ≤ and ≥.

Now solve 3x − 4 < 11. Treat it like an equation and do the same to both sides. Adding 4 to both sides gives 3x < 15. A balance shows this. Say a bag weighs 4 kg. Then 3 bags weigh 12 kg, and the other pan holds 15 kg. The 15 kg side is heavier, so the beam tilts.

A balance that is not level. The left pan holds 3 bags marked x and is labelled 3x. The right pan holds 15 weights and is labelled 15. The right pan is lower, so it is heavier.xxx3x15
With x = 4, 3x is less than 15. The beam tilts, and it stays tilted if you take the same from both sides. Each small square is 1 kg.

Worked example

Solve 3x − 4 < 11 and show it on a number line.

  1. Add 4 to both sides. The sign stays the same: 3x < 15.
  2. Divide both sides by 3. The sign stays the same: x < 5.
  3. Test it. When x = 4, 3x − 4 = 8, which is less than 11. When x = 6, 3x − 4 = 14, which is not.
  4. On the number line put an open circle at 5 with a bar going left.

Answer: x < 5

A table with the head x, 3x − 4, < 11?. Rows: 4, 8, yes; 5, 11, no; 6, 14, no.x3x − 4< 11?48yes511no614no
x = 5 gives exactly 11, which is not less than 11. So 5 is left out and the circle is open.

Now a trap. On a number line the bigger number is always further right. 2 is left of 5, so 2 < 5. Multiply both numbers by −1 and look again.

A number line from −5 to 5 with the whole numbers marked. Dots mark 2 and 5, and also −2 and −5. 2 is left of 5, but −2 is right of −5.−5−4−3−2−1012345−5−225
2 < 5, but after multiplying by −1 the order turns round: −2 > −5.

Multiplying by −1 swaps left and right. So multiplying or dividing both sides by a negative number must flip the sign: < becomes > and > becomes <. Adding or taking away never flips it. Neither does multiplying or dividing by a positive number.

Worked example

Solve 7 − 2x > 1.

  1. Take 7 from both sides. The sign stays the same: −2x > −6.
  2. Divide both sides by −2. The number is negative, so the sign flips: x < 3.
  3. Test it. When x = 0, 7 − 2x = 7, which is more than 1. When x = 4, 7 − 2x = −1, which is not.

Answer: x < 3

A number line from −1 to 8. An open circle at 3 with a thick bar going left to an arrow. The label x is at the right end.−1012345678x
x < 3: the open circle leaves 3 out.

Sometimes x is boxed in on both sides. The inequality −2 < x ≤ 3 means x is more than −2 and also at most 3. The circle at −2 is open, and the circle at 3 is filled.

A number line from −4 to 5. An open circle at −2 and a filled circle at 3, with a thick bar between them. Dots mark the whole numbers −1, 0, 1, 2 and 3. The label x is at the right end.−4−3−2−1012345x
Here −2 < x ≤ 3, and the whole numbers that fit are the five dots.

Worked example

List the whole numbers that fit −2 < x ≤ 3.

  1. Read it as two rules. x is more than −2, and x is at most 3.
  2. The circle at −2 is open, so −2 does not fit. The first whole number is −1.
  3. The circle at 3 is filled, so 3 does fit. The last whole number is 3.
  4. The whole numbers are −1, 0, 1, 2, 3.
  5. Count them. There are 5 whole numbers.

Answer: −1, 0, 1, 2, 3

See it three ways

With things

A shelf holds fewer than 5 books. It might hold none. How many books could be on it?

As a picture

A number line from −1 to 7. A filled circle at 0 and an open circle at 5, with a thick bar between them. Dots mark 0, 1, 2, 3 and 4. The label x is at the right end.−101234567x

In words

The shelf holds 0, 1, 2, 3 or 4 books. It cannot hold 5, because 5 is not fewer than 5.

In symbols

x is a whole number with 0 ≤ x < 5.

Try it

Mark a rule with stones

You need: paper and a pencil, a few small stones or coins.

  1. Draw a number line from 0 to 8 and write the numbers under it.
  2. The rule is x ≥ 3. Put a stone on every whole number that fits.
  3. Mark 3 with a filled circle and draw a bar with an arrow to the right.

What you will see: The stones sit on 3, 4, 5, 6, 7 and 8. The number 3 has a stone because ≥ includes it, so the circle is filled. The arrow shows that 9, 10 and every bigger number fit too.

If you cannot try it, this is what happens.
A number line from 0 to 8. A filled circle at 3 with a thick bar going right to an arrow. Dots mark 3, 4, 5, 6, 7 and 8. The label x is at the right end.012345678x

A wrong idea many people have

You flip the sign whenever you divide.

Why it feels right: Flipping is the new rule, so it feels as if it applies to every division.

The test that breaks it: Solve 3x < 15. Dividing by 3 gives x < 5. Put x = 4: 3 × 4 = 12, which is less than 15. A flipped sign, x > 5, would say 4 is not allowed.

What is true: The sign flips only when you multiply or divide both sides by a negative number. Dividing by 3 is dividing by a positive number.

A wrong idea many people have

x > 3 and x ≥ 3 mean the same.

Why it feels right: The two signs look almost the same, and both point to the bigger numbers.

The test that breaks it: Put x = 3. Then 3 ≥ 3 is true, because 3 equals 3. But 3 > 3 is false.

What is true: x > 3 leaves 3 out, with an open circle. x ≥ 3 keeps 3, with a filled circle.

Key words

inequality
a statement that one thing is less than, more than, at most or at least another. Two examples are x < 5 and x ≥ 3.
open circle
a ring on a number line. It shows the end number is not included.
filled circle
a solid dot on a number line. It shows the end number is included.

Back to your guess

The answer is C: x < 5, and every number below 5 fits.

An inequality has many answers. 3x − 4 < 11 gives 3x < 15, so x < 5. Every number below 5 fits. The sign flips only when you multiply or divide both sides by a negative number.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which inequality does the number line show?

    A number line from 0 to 8. An open circle at 4 with a thick bar going right to an arrow. The label x is at the right end.012345678x
    Tap your answer.

    1. Not this one. The bar goes towards the bigger numbers, to the right. Less than would go to the left. Go to M4.8.3. Try another.

    2. Yes, that is right. The bar goes right, so x is more than 4. The circle is open, so 4 itself is not included.

    3. Not this one. A filled circle would show that 4 is included. This circle is open, so 4 is left out. Go to M4.8.3. Try another.

    See the answer

    x > 4

    The bar goes right, so x is more than 4. The circle is open, so 4 itself is not included.

    • If you chose A: The bar goes towards the bigger numbers, to the right. Less than would go to the left. Go to M4.8.3.
    • If you chose C: A filled circle would show that 4 is included. This circle is open, so 4 is left out. Go to M4.8.3.

  2. Solve 2x + 3 < 11.

    Tap your answer.

    1. Not this one. You added 3 to the 11. The 3 is added on the left, so take it from both sides instead. Go to M4.8.3. Try another.

    2. Not this one. Taking 3 from 11 gives 2x < 8, which is right. But 2x < 8 still needs dividing by 2. Go to M4.8.3. Try another.

    3. Not this one. The sign flips only when you multiply or divide by a negative number. Dividing by 2 keeps the sign. Go to M4.8.3. Try another.

    4. Yes, that is right. Take 3 from both sides: 2x < 8. Divide both sides by 2: x < 4. The sign stays because 2 is positive.

    See the answer

    x < 4

    Take 3 from both sides: 2x < 8. Divide both sides by 2: x < 4. The sign stays because 2 is positive.

    • If you chose A: You added 3 to the 11. The 3 is added on the left, so take it from both sides instead. Go to M4.8.3.
    • If you chose B: Taking 3 from 11 gives 2x < 8, which is right. But 2x < 8 still needs dividing by 2. Go to M4.8.3.
    • If you chose C: The sign flips only when you multiply or divide by a negative number. Dividing by 2 keeps the sign. Go to M4.8.3.

  3. Solve 9 − 2x < 3.

    Tap your answer.

    1. Yes, that is right. Take 9 from both sides: −2x < −6. Divide both sides by −2 and flip the sign: x > 3. Test x = 4: 9 − 8 = 1, which is less than 3.

    2. Not this one. You divided both sides by −2 but did not flip the sign. Dividing by a negative number turns < into >. Go to M4.8.3. Try another.

    3. Not this one. You flipped the sign, which is right. But −6 ÷ −2 is +3, because a negative divided by a negative is positive. Go to M4.8.3. Try another.

    See the answer

    x > 3

    Take 9 from both sides: −2x < −6. Divide both sides by −2 and flip the sign: x > 3. Test x = 4: 9 − 8 = 1, which is less than 3.

    • If you chose B: You divided both sides by −2 but did not flip the sign. Dividing by a negative number turns < into >. Go to M4.8.3.
    • If you chose C: You flipped the sign, which is right. But −6 ÷ −2 is +3, because a negative divided by a negative is positive. Go to M4.8.3.

  4. Which list holds all the whole numbers that fit −3 < x ≤ 2?

    A number line from −4 to 4. An open circle at −3 and a filled circle at 2, with a thick bar between them. The label x is at the right end.−4−3−2−101234x
    Tap your answer.

    1. Not this one. The circle at −3 is open, so −3 is not included. The list must start at −2. Go to M4.8.3. Try another.

    2. Not this one. The circle at 2 is filled, so 2 is included. The list must end at 2. Go to M4.8.3. Try another.

    3. Yes, that is right. x is more than −3, so −3 is out and −2 is the first. Also x is at most 2, so 2 is in. That gives five whole numbers.

    4. Not this one. This includes −3, which has an open circle, and leaves out 2, which has a filled circle. The ends are the wrong way round. Go to M4.8.3. Try another.

    See the answer

    −2, −1, 0, 1, 2

    x is more than −3, so −3 is out and −2 is the first. Also x is at most 2, so 2 is in. That gives five whole numbers.

    • If you chose A: The circle at −3 is open, so −3 is not included. The list must start at −2. Go to M4.8.3.
    • If you chose B: The circle at 2 is filled, so 2 is included. The list must end at 2. Go to M4.8.3.
    • If you chose D: This includes −3, which has an open circle, and leaves out 2, which has a filled circle. The ends are the wrong way round. Go to M4.8.3.

  5. A number line has a filled circle at 2 and a bar going left. What does it show?

    A number line from −2 to 6. A filled circle at 2 with a thick bar going left to an arrow. The label x is at the right end.−2−10123456x
    Tap your answer.

    1. Not this one. The circle is filled, not open. A filled circle means the end number is included. Go to M4.8.3. Try another.

    2. Not this one. The bar goes left, to the smaller numbers. More than or equal to would go right. Go to M4.8.3. Try another.

    3. Yes, that is right. The bar goes left, so x is at most 2. The circle is filled, so 2 itself is included.

    See the answer

    x ≤ 2, and the end point is included.

    The bar goes left, so x is at most 2. The circle is filled, so 2 itself is included.

    • If you chose A: The circle is filled, not open. A filled circle means the end number is included. Go to M4.8.3.
    • If you chose B: The bar goes left, to the smaller numbers. More than or equal to would go right. Go to M4.8.3.

  6. Which step needs the sign of the inequality to be flipped?

    Tap your answer.

    1. Not this one. The number 4 is positive. Dividing by a positive number keeps the sign the same. Go to M4.8.3. Try another.

    2. Yes, that is right. Only this step divides by a negative number. Dividing both sides of −4x < 20 by −4 gives x > −5, and the sign flips.

    3. Not this one. Adding or taking away never flips the sign. Only multiplying or dividing by a negative number does. Go to M4.8.3. Try another.

    4. Not this one. The number 2 is positive. Multiplying by a positive number keeps the sign the same. Go to M4.8.3. Try another.

    See the answer

    −4x < 20, divided by −4 on both sides

    Only this step divides by a negative number. Dividing both sides of −4x < 20 by −4 gives x > −5, and the sign flips.

    • If you chose A: The number 4 is positive. Dividing by a positive number keeps the sign the same. Go to M4.8.3.
    • If you chose C: Adding or taking away never flips the sign. Only multiplying or dividing by a negative number does. Go to M4.8.3.
    • If you chose D: The number 2 is positive. Multiplying by a positive number keeps the sign the same. Go to M4.8.3.

  7. Solve 5x + 2 < 17.

    Tap your answer.

    1. Yes, that is right. Take 2 from both sides: 5x < 15. Divide both sides by 5. The sign stays, because 5 is positive: x < 3.

    2. Not this one. The sign flips only when you multiply or divide both sides by a negative number. Here you divide by 5, which is positive, so the sign stays. Go to M4.8.3. Try another.

    3. Not this one. Taking 2 from both sides gives 5x < 15, which is right. But 5 still multiplies x, so divide both sides by 5 as well. Go to M4.8.3. Try another.

    4. Not this one. You added 2 to the 17. The 2 is added to 5x, so take it from both sides: 5x < 15. Go to M4.8.3. Try another.

    See the answer

    x < 3

    Take 2 from both sides: 5x < 15. Divide both sides by 5. The sign stays, because 5 is positive: x < 3.

    • If you chose B: The sign flips only when you multiply or divide both sides by a negative number. Here you divide by 5, which is positive, so the sign stays. Go to M4.8.3.
    • If you chose C: Taking 2 from both sides gives 5x < 15, which is right. But 5 still multiplies x, so divide both sides by 5 as well. Go to M4.8.3.
    • If you chose D: You added 2 to the 17. The 2 is added to 5x, so take it from both sides: 5x < 15. Go to M4.8.3.

  8. Solve 14 − 2x < 6.

    Tap your answer.

    1. Not this one. You divided both sides by −2 but kept the sign. Dividing by a negative number turns < into >. Go to M4.8.3. Try another.

    2. Not this one. You flipped the sign, which is right. But −8 ÷ −2 is +4, because a negative divided by a negative is positive. Go to M4.8.3. Try another.

    3. Not this one. Taking 14 from both sides gives −2x < −8, which is right. But −2 still multiplies x, so divide both sides by −2 as well. Go to M4.8.3. Try another.

    4. Yes, that is right. Take 14 from both sides: −2x < −8. Divide both sides by −2 and flip the sign: x > 4. Test x = 5: 14 − 10 = 4, which is less than 6.

    See the answer

    x > 4

    Take 14 from both sides: −2x < −8. Divide both sides by −2 and flip the sign: x > 4. Test x = 5: 14 − 10 = 4, which is less than 6.

    • If you chose A: You divided both sides by −2 but kept the sign. Dividing by a negative number turns < into >. Go to M4.8.3.
    • If you chose B: You flipped the sign, which is right. But −8 ÷ −2 is +4, because a negative divided by a negative is positive. Go to M4.8.3.
    • If you chose C: Taking 14 from both sides gives −2x < −8, which is right. But −2 still multiplies x, so divide both sides by −2 as well. Go to M4.8.3.

  9. Solve 4x + 5 < 29. What number must x be less than?

    See the answer

    6

    Take 5 from both sides: 4x < 24. Divide both sides by 4. The sign stays, so x < 6.

  10. Solve 19 − 3x < 4. What number must x be more than?

    See the answer

    5

    Take 19 from both sides: −3x < −15. Divide both sides by −3 and flip the sign, so x > 5.

  11. From an earlier chapter

    Use t = (v − u)/a with v = 41, u = 5 and a = 9. What is t?

    See the answer

    t = 4

    Put the numbers in: t = (41 − 5)/9 = 36/9 = 4.

Keep it

Copy this into your exercise book.

Open circle: not included. Filled circle: included. Solve like an equation. Flip the sign only when you multiply or divide by a negative number.

Draw: A number line with an open circle at 5 and a bar going left, and the words x < 5.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (989 KB). It opens on any phone.