Inequalities

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Notes de leçon Diapos Questions d'entraînement

Notes de leçon

Inequality symbols

< less than, > greater than, \leq less than or equal to, \geq greater than or equal to.
-2 < x \leq 3 means x is greater than -2 and at most 3. Integers satisfying it: -1, 0, 1, 2, 3.

Solving linear inequalities

Solve like an equation, doing the same to both sides. The answer is a range: 3x + 2 > 11 gives x > 3.
Reverse the inequality sign if you multiply or divide by a negative number: -2x < 8 gives x > -4.

Number lines and set notation

A filled circle shows the end value is included (\leq, \geq); an open circle shows it is excluded (<, >).
Write a double inequality such as -1 \leq x < 4 as one range on a number line.

Quadratic inequalities

Solve x² - x - 6 < 0: factorise (x - 3)(x + 2) < 0. Roots -2 and 3. The parabola is below the axis between the roots, so -2 < x < 3.
For x² > 9: x < -3 or x > 3.

Where the quadratic is negative

Graphical inequalities

Draw a solid boundary for ≤ or ≥, and a dashed boundary for < or >. Test a point not on the boundary to choose the required side; the origin works only if it is off the boundary.
Regions defined by several inequalities: shade out the unwanted regions and leave the required region unshaded, or as the question states.

Diapos

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Questions d'entraînement

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1.Which symbol means "less than or equal to"?
Easy
- A<
- B\leq
- C>
- D\geq
2.Solve x + 4 < 10.
Easy
- Ax < 6
- Bx > 6
- Cx < 14
- Dx > 14
3.Solve 2x \geq 8.
Easy
- Ax \geq 4
- Bx \leq 4
- Cx \geq 16
- Dx \geq 6
4.Which integers satisfy -2 < x \leq 1?
Easy
- A-1, 0, 1
- B-2, -1, 0, 1
- C-1, 0
- D-2, -1, 0
5.Solve 3x - 5 \leq 10.
Easy
- Ax \leq 5
- Bx \geq 5
- Cx \leq 15
- Dx \leq 1.67
6.Solve -2x < 8.
Easy
- Ax < -4
- Bx > -4
- Cx < 4
- Dx > 4
7.Solve 5 - x > 2.
Easy
- Ax > 3
- Bx < 3
- Cx > -3
- Dx < -3
8.What does an open circle on a number line mean?
Easy
- Athe value is included
- Bthe value is not included
- Cthe value is negative
- Dthere is no solution

Plus de questions

9.What does a filled circle on a number line mean?
Easy
- Athe value is included
- Bthe value is excluded
- Cmultiplied by 2
- Dno solution
10.Solve 2(x - 3) < 10.
Medium
- Ax < 8
- Bx < 5
- Cx > 8
- Dx < 13
11.Solve 4x + 1 \geq 7x - 8.
Medium
- Ax \leq 3
- Bx \geq 3
- Cx \leq -3
- Dx \geq 9
12.Solve -3 \leq 2x + 1 < 7.
Medium
- A-2 \leq x < 3
- B-4 \leq x < 6
- C-2 \leq x < 4
- D-1 \leq x < 3
13.How many integers satisfy -3 \leq 2x + 1 < 7?
Medium
- A4
- B5
- C6
- D3
14.The largest integer satisfying 3x - 2 < 13 is:
Medium
- A4
- B5
- C6
- D3
15.Solve x² < 16.
Medium
- Ax < 4
- B-4 < x < 4
- Cx > 4 or x < -4
- Dx < -4
16.Solve x² - x - 6 < 0.
Medium
- A-2 < x < 3
- Bx < -2 or x > 3
- C-3 < x < 2
- Dx < 3
17.Solve x² > 9.
Medium
- Ax > 3
- B-3 < x < 3
- Cx < -3 or x > 3
- Dx > -3
18.Solve x² - 5x + 4 \geq 0.
Medium
- A1 \leq x \leq 4
- Bx \leq 1 or x \geq 4
- Cx \geq 4
- Dx \leq 4
19.What is the largest integer n for which n² < 50?
Medium
- A7
- B8
- C6
- D25
20.On a graph, the boundary for y > 2x is:
Easy
- Asolid
- Bdashed
- Cdotted vertical
- Dabsent
21.On a graph, the boundary for y \leq 3 is:
Easy
- Asolid
- Bdashed
- Cabsent
- Dcurved
22.Which listed point can be used to test a side of the boundary y = x + 1 for the inequality y < x + 1?
Medium
- A(0, 0)
- B(1, 2), on the boundary
- CNo test point is needed
- D(2, 3), on the boundary
23.Which inequality describes the region below the line y = 4, including the line?
Medium
- Ay < 4
- By \leq 4
- Cy > 4
- Dy \geq 4
24.Which point satisfies y > x and y < 5?
Medium
- A(1, 3)
- B(3, 1)
- C(2, 5)
- D(6, 4)
25.Which inequality means "x is at least 5"?
Easy
- Ax > 5
- Bx \geq 5
- Cx < 5
- Dx \leq 5
26.Which inequality means "x is less than 10"?
Easy
- Ax \leq 10
- Bx < 10
- Cx > 10
- Dx \geq 10
27.When multiplying or dividing an inequality by a negative number, the sign flips.
Easy
True or false?
28.x \leq 3 is shown with an open circle.
Easy
True or false?
29.x² > 4 means x > 2 only.
Easy
True or false?
30.The solution of x² < 4 is -2 < x < 2.
Easy
True or false?
31.-3x > 6 gives x > -2.
Medium
True or false?
32.Which of these integers satisfy 2 < x \leq 6?
Medium
- A2
- B3
- C4
- D6
- E7
33.Which values satisfy x \geq 3?
Medium
- A3
- B2.5
- C4
- D10
- E2.99
34.Solve x² - x - 6 < 0 and write the solution as an inequality.
Medium

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