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Inequalities for AQA A-level Maths

Inequalities

This page covers the following topics:

1. Quadratic inequalities
2. Direction of inequalities

To solve quadratic inequalities, rearrange them so that one side is 0. Then solve the corresponding quadratic equation. Sketch the graph of the quadratic equation using the roots and deduce the range of values the variable can take. When the quadratic expression is greater than 0, the wanted region is above the x-axis, and when the quadratic expression is less than 0, the wanted region is below the x-axis.

Quadratic inequalities

When solving inequalities, if it is required to multiply or divide by a negative number, the direction of the inequality sign must be reversed. When there is an expression in terms of the variable that is being solved in the denominator, multiply every term by the square of the expression, so that there is no need to reverse the inequality sign, as this will guarantee that the number that the terms are being multiplied by is positive.

Direction of inequalities

1

What is the set of values of x for which 4(xยฒ โˆ’ 2x) โ‰ค x(5 + 2x) + 7?

[โˆ’0.5, 7]

What is the set of values of x for which 4(xยฒ โˆ’ 2x) โ‰ค x(5 + 2x) + 7?

2

Solve โˆ’4x + 8 โ‰ค 56.

Subtracting 8 from both sides, โˆ’4x โ‰ค 48.
Dividing both sides by โˆ’4 and reversing the inequality sign gives x โ‰ฅ โˆ’12.

x โ‰ฅ โˆ’12

Solve โˆ’4x + 8 โ‰ค 56.

3

A painter wants to make a painting of sides x โˆ’ 7 and x โˆ’ 11. Given that he wants the area of the painting to be at least 12 unitsยฒ, find the range of possible values of x.

image

A painter wants to make a painting of sides x โˆ’ 7 and x โˆ’ 11. Given that he wants the area of the painting to be at least 12 unitsยฒ, find the range of possible values of x.

4

Solve 7(11 + x โˆ’ xยฒ) โ‰ค 5(x โˆ’ xยฒ) โˆ’ x.

image

Solve 7(11 + x โˆ’ xยฒ) โ‰ค 5(x โˆ’ xยฒ) โˆ’ x.

5

Find the range of possible values of x for 3/x โˆ’ 8 โ‰ค 4.

image

Find the range of possible values of x for 3/x โˆ’ 8 โ‰ค 4.

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