Inequalities flashcards
10 cards on Inequalities: the key points, worked examples and the slips to watch. Swipe through them, or print them.
Card 1 of 10
Inequalities
Inequality symbols
What do the symbols <, >, ≤ and ≥ mean, and why does an inequality have many solutions?
Answer it in your head, then turn the card over.
Inequality symbols
Key points
- < is less than and > is greater than: the end value itself is not included.
- ≤ is less than or equal to and ≥ is greater than or equal to: the end value is included.
- An inequality's solution is a range of values, not one number: x > 4 is true for 4.1, 7 and 100.
Watch out
Read the sign from the letter's side: in 5 > x, x is less than 5, so it is the same as x < 5.
Inequalities
Inequalities on a number line
How do you show an inequality on a number line?
Answer it in your head, then turn the card over.
Inequalities on a number line
Key points
- An open circle means the end value is not included: use it for < or >.
- A filled circle means the end value is included: use it for ≤ or ≥.
- Draw an arrow for a range that keeps going, or a line between two circles for a range with two ends.
- −3 < x ≤ 2 is an open circle at −3, a filled circle at 2, and a line joining them.
Watch out
x > 2 has an open circle at 2: a filled circle would mean 2 itself counts.
Inequalities
Solving a linear inequality
How do you solve an inequality like 4x − 3 < 13?
Answer it in your head, then turn the card over.
Solving a linear inequality
Key points
- Solve it like an equation: do the same to both sides.
- Keep the inequality sign on every line, so the answer is a range.
- Check by testing a value from your range in the original inequality.
4x − 3 < 13
Add 3: 4x < 16
Divide by 4: x < 4
Test x = 0: 4 × 0 − 3 = −3, and −3 < 13 holds
Watch out
Ending with x = 4 swaps the inequality for an equals sign: the answer is x < 4, and 4 itself does not work.
Inequalities
Dividing by a negative number
What happens to the inequality sign when you multiply or divide both sides by a negative number?
Answer it in your head, then turn the card over.
Dividing by a negative number
Key points
- Multiplying or dividing both sides by a negative number flips the sign: < becomes >.
- It happens because negatives reverse the order of numbers: 2 < 3, but −2 > −3.
- Adding, subtracting, or multiplying and dividing by a positive never flips it.
−5x < 20
Divide by −5 and flip the sign: x > −4
Test x = 0: −5 × 0 = 0, and 0 < 20 holds, as does 0 > −4
Watch out
Keeping the sign after dividing by a negative is the classic slip: −2x < 6 gives x > −3, not x < −3.
Inequalities
Listing integer solutions
How do you list the integers that satisfy an inequality?
Answer it in your head, then turn the card over.
Listing integer solutions
Key points
- Integers are whole numbers, including 0 and the negatives.
- Solve the inequality first, then list the whole numbers in the range.
- A strict < or > leaves the end out; ≤ or ≥ lets it in. For −3 < x ≤ 2: −2, −1, 0, 1, 2.
Positive integers with 3x − 1 < 11
Add 1: 3x < 12
Divide by 3: x < 4
So x = 1, 2 or 3 (4 is left out)
Watch out
Do not include the end of a strict inequality: the list for −3 < x ≤ 2 starts at −2, not −3.
Inequalities
Double inequalities
How do you solve a double inequality like −5 ≤ 2x + 1 < 7?
Answer it in your head, then turn the card over.
Double inequalities
Key points
- Do the same thing to all three parts at once.
- Get x on its own in the middle, keeping both signs.
- Then read off the integers if the question asks for them.
−5 ≤ 2x + 1 < 7
Take 1 from each part: −6 ≤ 2x < 6
Divide each part by 2: −3 ≤ x < 3
Integers: −3, −2, −1, 0, 1, 2
Watch out
Each step must reach every part: taking 1 from the middle only leaves −5 ≤ 2x < 7, which is not the same.
Inequalities
HigherQuadratic inequalities: less than
How do you solve a quadratic inequality like x² − x − 6 < 0?
Answer it in your head, then turn the card over.
Quadratic inequalities: less than
Key points
- Solve the matching equation first to find the roots.
- Sketch the U-shaped curve through the roots on the x-axis.
- Less than 0 means below the x-axis: between the roots, written as one double inequality.
x² − x − 6 < 0
(x + 2)(x − 3) = 0, so the roots are −2 and 3
The U dips below the axis between them
So −2 < x < 3 (test x = 0: 0 − 0 − 6 = −6, below 0)
Watch out
x² < 9 is not just x < 3: −4 would fit that, but (−4)² = 16. The answer is −3 < x < 3.
Inequalities
HigherQuadratic inequalities: greater than
How do you write the answer when a quadratic inequality says greater than 0?
Answer it in your head, then turn the card over.
Quadratic inequalities: greater than
Key points
- Greater than 0 means above the x-axis: the two outside regions of the U.
- So the answer is two separate inequalities joined by "or".
- Move every term to one side first, so the other side is 0.
x² > 9
x² − 9 > 0, and x² − 9 = 0 gives x = −3 or x = 3
Above the axis outside the roots: x < −3 or x > 3
Test x = 4: 4² = 16, and 16 > 9 holds
Watch out
Never write the two outside parts as one inequality: 3 < x < −3 cannot be true for any x.
Inequalities
HigherBoundary lines for regions
How do you draw the boundary line for an inequality like y < 2x + 1 on a graph?
Answer it in your head, then turn the card over.
Boundary lines for regions
Key points
- Draw the line you get by swapping the sign for =: here y = 2x + 1.
- Use a dashed line for < or >, because points on the line are not included.
- Use a solid line for ≤ or ≥, because points on the line count.
- x = a is a vertical line; y = b is a horizontal one.
y < 2x + 1
Points on y = 2x + 1: x = 0 gives y = 1, x = 1 gives y = 3
Join them with a dashed line, because < is strict.
Watch out
A dashed line is the graph's open circle: a solid line would put the points on the line in the region.
Inequalities
HigherShading the region
How do you decide which side of a boundary line to shade, and which region is the answer?
Answer it in your head, then turn the card over.
Shading the region
Key points
- Test a point not on the line, usually (0, 0): put its x and y into the inequality.
- If the statement is true, that side is in the region; if not, it is the other side.
- With several inequalities, the answer is the region where all of them hold. Label it R if asked.
Which side of y = 2x + 1 is y < 2x + 1?
Test (0, 0): is 0 < 2 × 0 + 1 = 1? Yes.
So the region is on the side containing (0, 0).
Watch out
If the line passes through (0, 0), that point cannot decide it: test another, such as (1, 0).
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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