A question can ask to find a set of values which satisfy an inequality equation. If the equation is a quadratic expression, then you should always sketch a graph in order to achieve your answer.

The graph above shows a small sketch of y = (x – 1) (x + 5)

**Important Rule:**

The red shaded area shows one set of values for x, the curve is below the x-axis, so y < 0

The green shaded areas show another set of values for x, the curve is above the x-axis, so y > 0.

There are a total of two separate set of values that satisfy this inequality. You will need to mention both these set of values and write ‘or’ between them.

**Quadratic Inequality Steps:**

1) **Rearrange** the equation so **one side is equal to 0**

2) **Factorise** the **other side** of the equation

3) **Proceed** to sketch the graph

4) Write down the **solutions** using the **appropriate inequality symbol**

**Simple Inequality Question: (Basically solving a normal equation)**

Find the set values for: (2 marks)

8x – 7 < 5x + 5

-5x

3x -7 < 5

+7

3x < 12

12 / 3 = 4

X < 4

**Inequality Question with Sketched Graph: **

Find the set of values for which:

2x** ^{2}** – 5x – 3 > 0 (0 is already on one side so no need to rearrange)

(2x + 1) (x – 3) (Factorising the equation above)

X = -0.5 or x = 3 (Finding the roots in order to draw graph)

Draw the graph:

Answer: X < -0.5 or x < 3

**Find the set values of both:**

8x – 7 < 5x + 5 and 2x** ^{2}** – 5x – 3 > 0

As you have already worked out the values of each individual equation, you can use an inequality line graph to determine the set of values for both of these inequalities:

This shows that:

X < =0.5 or 3 < x < 4

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