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13.6 Inequalities

3 N is an integer. Work out:
a the smallest possible value of N if N ≥ 6.5
b the largest possible value of N if N < −3
c the possible values of N if N ≥ −2 and N < 2

4 Solve these inequalities.
a 5x > 7 b 4x + 1 ≤ 15 c 3x + 1 < −6 d 3(x + 1) ≥ −6

5 Show each solution set in question 4 on a number line.
6 You are given that z > 2.
Write an inequality for each expression.
a 2z + 9 b 3(z − 4) c 4 + 2z d 5(3z − 2)
7 Solve these inequalities.
a 2(a + 4) < 15 b 3b − 4 ≥ b + 18 c c + 18 ≤ 30 − c d 3(d + 5) > 2(d − 6)
8 The perimeter of this triangle is not more than 30 cm.
a Write an inequality to show this. 2n + 3 cm
b Solve the inequality. n cm
c What are the largest possible lengths of the sides?
2n + 2 cm

9 The diagram shows four angles round a point. x°
a Write an inequality for x. 2x°
b Solve the inequality. x + 30°
c Explain why the angle labelled x° cannot be a right angle.

Summary

You should now know that: You should be able to:
+ Linear equations can be solved by algebraic + Construct and solve linear equations with integer
manipulation, doing the same thing to each side coefficients (with and without brackets, negative
of the equation. signs anywhere in the equation, positive or
+ Number problems can be solved by setting up negative solution); solve a number problem by
equations and solving them. constructing and solving a linear equation.
+ Two equations with two unknowns are called + Solve a simple pair of simultaneous linear
simultaneous equations. They can be solved by equations by eliminating one variable.
eliminating one variable. + Understand and use inequality signs (<, >, ≤, ≥);

+ Some equations cannot easily be solved by construct and solve linear inequalities in one
algebraic manipulation. Solutions can be found by variable; represent the solution set on a number
systematic trial and improvement. line.
+ Linear inequalities can be solved in a similar way + Use systematic trial and improvement methods to
to linear equations. find approximate solutions of equations such as
x + 2x = 20.
2
+ Manipulate numbers, algebraic expressions and
equations, and apply routine algorithms.
+ Check results by using inverse operations.

13 Equations and inequalities 133