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Inequalities on the number line

Algebra · level 4, lesson 6 of 7

A question to start

Start with a true statement: \(-2 < 3\). Now multiply both sides by \(-1\). You get \(2\) and \(-3\). Is it still true that \(2 < -3\)?

No! 2 is bigger than \(-3\). The correct statement is \(2 > -3\). Multiplying both sides by a negative number turned the inequality sign around.

With equations, you can do anything to both sides and the equation stays true. With inequalities, almost everything works the same way, with one famous exception: multiplying or dividing by a negative number flips the sign. This lesson shows how to solve inequalities, how to draw their answers, and why the flip happens.

The idea

Inequalities have many solutions

An inequality compares two amounts using one of four signs:

Sign Read as
\(<\) is less than
\(>\) is greater than
\(\le\) is less than or equal to
\(\ge\) is greater than or equal to

In level 3 you listed whole numbers that make \(n + 3 < 8\) true. Now we allow all numbers: fractions, decimals and negatives. Then \(x + 4 > 9\) is true for \(x = 6\), \(x = 5.1\), \(x = 5.001\), \(x = 100\), and for every number bigger than 5. That is infinitely many numbers, so we cannot list them. Instead we write the answer as a short inequality, \(x > 5\), and draw it.

Drawing on a number line

To draw \(x > 5\), put a dot at 5 and shade (or draw an arrow along) the part of the line where the answers are.

  • An open dot (a hollow circle) means the edge number is not included. Use it for \(<\) and \(>\).
  • A closed dot (a filled circle) means the edge number is included. Use it for \(\le\) and \(\ge\).
0123 45678 x > 5 (open dot: 5 is not included) −4−3−2−1 01234 x ≤ −1 (closed dot: −1 is included)

Solving: the same as equations, almost

These steps keep an inequality true, just like with an equation:

  • adding the same number to both sides;
  • subtracting the same number from both sides;
  • multiplying or dividing both sides by the same positive number.

Why? Think of two people on a number line, the one on the left being smaller. If both walk 4 steps right (add 4), or both walk 4 steps left (subtract 4), the left one is still on the left. If both double their distance from 0 (multiply by 2), the order stays the same too: \(3 < 5\) becomes \(6 < 10\).

So \(x + 4 > 9\) is solved like an equation: subtract 4 from both sides, \(x > 5\).

Why a negative flips the sign

Multiplying by \(-1\) changes every number into its opposite: 3 becomes \(-3\), \(-2\) becomes 2. On a number line, that is a reflection in 0: everything swaps sides, like a mirror placed at zero.

−3−2−10 123 −2 becomes 2 3 becomes −3

Before the reflection, \(-2\) was on the left of 3. After it, 2 is on the right of \(-3\). A mirror swaps left and right, so the smaller number becomes the bigger one. That is why

\[-2 < 3 \quad\text{becomes}\quad 2 > -3.\]

Multiplying by any negative number does the same thing, because it is a reflection (multiply by \(-1\)) together with a stretch (multiply by a positive number), and only the reflection changes the order. Dividing by a negative is the same as multiplying by a negative (dividing by \(-2\) is multiplying by \(-\tfrac{1}{2}\)), so it flips the sign too.

Rule: when you multiply or divide both sides of an inequality by a negative number, reverse the sign.

A way to avoid the flip

You can always avoid dividing by a negative by moving the letter term to the other side first. For \(-3x \ge 12\), add \(3x\) to both sides and subtract 12: \(-12 \ge 3x\). Divide by 3 (positive, no flip): \(-4 \ge x\), which means \(x \le -4\). The same answer as dividing by \(-3\) and flipping. Both ways are correct; choose the one you trust.

Checking an inequality

Check two things:

  1. The edge. Put the edge number in: the two sides should be equal.
  2. The direction. Put in one number that should work and one that should not.

Worked examples

1. One step. Solve \(x + 4 > 9\) and draw it.

  • Subtract 4 from both sides: \(x > 5\).
  • Draw an open dot at 5 and an arrow to the right (the top line in the first drawing).
  • Check: edge \(5 + 4 = 9\) ✓; try 6: \(10 > 9\) ✓; try 0: \(4 > 9\) false ✓.

2. Two steps. Solve \(2x - 3 \le 7\).

  • Add 3: \(2x \le 10\). Divide by 2 (positive): \(x \le 5\).
  • Closed dot at 5, arrow to the left.
  • Check: \(x = 5\) gives \(7 \le 7\) ✓; \(x = 0\) gives \(-3 \le 7\) ✓; \(x = 6\) gives \(9 \le 7\) false ✓.

3. Dividing by a negative. Solve \(-3x \ge 12\).

  • Divide both sides by \(-3\) and flip: \(x \le -4\).
  • Check: \(x = -5\) gives \(15 \ge 12\) ✓; \(x = 0\) gives \(0 \ge 12\), false ✓. So the answers really are on the left of \(-4\).

4. The letter being subtracted. Solve \(10 - x < 4\).

  • Subtract 10: \(-x < -6\).
  • Multiply by \(-1\) and flip: \(x > 6\).
  • Check: \(x = 7\) gives \(3 < 4\) ✓; \(x = 6\) gives \(4 < 4\), false, so 6 is not included ✓.

5. A real problem. A lift can carry at most 600 kg. A man of 80 kg wants to load boxes of 40 kg each. How many boxes can he take with him?

  • Let \(b\) be the number of boxes. \(80 + 40b \le 600\).
  • Subtract 80: \(40b \le 520\). Divide by 40: \(b \le 13\).
  • He can take at most 13 boxes. Check: \(80 + 520 = 600\) ✓; 14 boxes would be 640 kg, too heavy.

Try it

  1. Write in symbols: (a) \(x\) is at least 3; (b) \(y\) is less than \(-2\). Draw each on a number line (say which dot).
  2. Solve \(x - 6 < 2\).
  3. Solve \(4x \ge 28\).
  4. Solve \(3x + 5 > 20\).
  5. Solve \(-2x < 8\).
  6. Solve \(7 - 2x \ge 1\).
  7. Is \(-5 < -1\)? Multiply both sides by \(-1\) and write the correct inequality.
  8. A phone plan costs 15 dollars a month plus 2 dollars per GB of data. You can spend at most 35 dollars. How many GB can you use?
Answers
  1. (a) \(x \ge 3\): closed dot at 3, arrow right. (b) \(y < -2\): open dot at \(-2\), arrow left.
  2. Add 6: \(x < 8\).
  3. Divide by 4: \(x \ge 7\).
  4. Subtract 5: \(3x > 15\). Divide by 3: \(x > 5\). Check: \(x = 6\) gives \(23 > 20\) ✓.
  5. Divide by \(-2\) and flip: \(x > -4\). Check: \(x = 0\) gives \(0 < 8\) ✓; \(x = -5\) gives \(10 < 8\), false ✓.
  6. Subtract 7: \(-2x \ge -6\). Divide by \(-2\) and flip: \(x \le 3\). Check: \(x = 3\) gives \(1 \ge 1\) ✓; \(x = 0\) gives \(7 \ge 1\) ✓; \(x = 4\) gives \(-1 \ge 1\), false ✓.
  7. Yes, \(-5 < -1\). Multiplying by \(-1\) gives 5 and 1, and \(5 > 1\): the sign flipped.
  8. \(15 + 2g \le 35\), so \(2g \le 20\) and \(g \le 10\). At most 10 GB.

Common mistake

Forgetting to flip. In \(-2x < 8\), dividing by \(-2\) without flipping gives \(x < -4\). Test \(x = -5\): \(10 < 8\) is false, so that answer is wrong. Testing one number catches this every time.

Flipping when you shouldn't. Subtracting a negative or adding a negative never flips the sign. Only multiplying or dividing by a negative does. In \(x - 3 > -7\), just add 3: \(x > -4\).

Where it leads