Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Lesson 3 of this unit. Solve 2x + 7 = 19.
Answer
x = 6. 19 − 7 = 12, then 12 ÷ 2 = 6. - From Unit 3 (integers), lesson 1. Which is greater, −3 or −8?
Answer
−3. It is closer to zero, so it is to the right on a number line. - From Lesson 4 of this unit. Solve 4x = x + 12.
Answer
x = 4. Take x from both sides: 3x = 12.
Words used in this lesson
- inequality
- a comparison using <, >, ≤ or ≥. For example: x > 4.
- at least
- equal to or more than. For example: at least 120 cm means x ≥ 120.
- at most
- equal to or less than. For example: at most 5 means x ≤ 5.
- boundary
- the number where the range starts. For example: 4 in x > 4.
- solution set
- all the numbers that make it true. For example: every number above 4.
Not one answer, but a range
A roller coaster in Sharjah has a sign: Riders must be at least 120 cm tall. A girl who is 118 cm cannot ride. One who is 135 cm can. What about a rider who is exactly 120 cm?
Predict first: which of the four symbols, <, ≤, > or ≥, matches “at least 120”? Is 120 itself a solution? Then change the symbol in the figure and see what moves.
x < 3: every number to the left of 3, but not 3 itself, so the circle is open. The test value 5 is not in the solution set.
Show me the steps
- x < 3: an open circle at 3 and an arrow to the left. The test value 5 is not less than 3, so it is not in the set.
- x ≤ 3: the circle at 3 is now filled, because 3 is included. The test value 3 would be a solution.
- x > 3: an open circle at 3 and an arrow to the right. The test value 5 is greater than 3, so it is in the set.
- x ≥ 3: a filled circle at 3 and an arrow to the right. Set b = −2 and the same shape sits at −2.
So “at least 120 cm” is x ≥ 120, with a filled circle at 120: the rider who is exactly 120 cm can ride, so the boundary is one of the solutions. “Taller than 120 cm” would be x > 120, with an open circle, and would leave 120 out.
An inequality compares two things with <, >, ≤ or ≥. Its solution is a range of numbers, drawn as a ray on a number line.
Open circle for < and >: the boundary is not included. Filled circle for ≤ and ≥: it is.
Solve it like an equation: do the same to both sides. Adding, subtracting, and multiplying or dividing by a positive number keep the sign as it is.
Solving keeps the solutions the same
Each solving step gives a new inequality with exactly the same solutions. The dots show it: a filled dot is a whole number that works.
3x + 5 ≤ 20: a filled dot marks each whole number that makes it true. Solve it one step at a time and watch the dots.
Show me the steps
- 3x + 5 ≤ 20: the whole numbers up to 5 are true (filled dots), and the ones above 5 are false.
- Subtract 5 from both sides: 3x ≤ 15. The same dots are filled.
- Divide both sides by 3, which is positive, so the sign stays: x ≤ 5. The same dots are filled, and the ray starts at 5 with a filled circle.
- Check: x = 4 gives 3 × 4 + 5 = 17, and 17 ≤ 20 is true.
- Inequality 3, 4x + 1 < 9: subtract 1 to get 4x < 8, divide by 4 to get x < 2. The circle at 2 is open: 4 × 2 + 1 = 9, which is not less than 9.
Two slips to watch for. The answer to 2x + 1 ≤ 9 is x ≤ 4, not x = 4: it is a range, so write it with the sign. And x > 4 does not include 4: at x = 4 the left side of 2x + 1 > 9 is exactly 9, and 9 is not greater than 9.
Worked examples
Solve x − 3 ≥ 5 and show it on a number line.
- Add 3 to both sides: x ≥ 8.
- The sign has an equals part, so the circle at 8 is filled; draw an arrow to the right.
- Check: x = 8 gives 5 ≥ 5, true; x = 7 gives 4 ≥ 5, false.
Solve 4x + 3 > 19.
- Subtract 3 from both sides: 4x > 16.
- Divide both sides by 4: x > 4. The circle at 4 is open; the arrow points right.
- Check: x = 5 gives 23 > 19, true; x = 4 gives 19 > 19, false.
Write “x is at least −2” as an inequality and draw it. At least means “equal to or more than”: x ≥ −2. A filled circle at −2, with an arrow to the right.
Your go. The first step is given. Solve 8x − 5 ≤ 19.
- Add 5 to both sides: 8x ≤ 24.
- Finish the solving, say which circle you draw, and then check one value.
Check the answer
Divide by 8: x ≤ 3, with a filled circle at 3 and an arrow to the left. Check: x = 3 gives 8 × 3 − 5 = 19, and 19 ≤ 19 is true.
Check before you go on
1. x ≥ 2 is drawn on a number line. Which description is right?
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. What is the smallest whole number x with x > 7?
3. x ≤ 4 and x is a positive whole number. How many values can x take?
Level 2 · Building
4. Solve x + 6 < 10. What is the largest whole number x can be?
5. Solve 3x − 3 ≥ 12. Which is the solution?
6. Which inequality has the solution x < 3?
Level 3 · Stretch
7. A phone plan costs 15 dirhams a month plus 2 dirhams for each text. Sara has 75 dirhams. What is the largest number of texts she can send?
8. Solve 5x + 2 > 3x + 12. What is the smallest whole number x can be?
Something to think about
Is there any number that satisfies both x < 3 and x > 5? Then draw “x is greater than −2 and at most 4” on one number line. How many whole numbers does it contain?
One way to think about it
No number is both below 3 and above 5, so the first has no solutions. The second has an open circle at −2, a filled circle at 4 and the line between them: −1, 0, 1, 2, 3 and 4, which is six whole numbers.
Exit ticket
9. Which describes the solution of 2x − 5 ≤ 7 on a number line?