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# Inequalities

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## Inequalities

4x - 5 < x + 1 is an inequality because the solution to the equation is a range of values of x. You can find the range of values of x, by solving the inequality as if it was a normal equation.

**For example:**

4x - 5 < x + 1 |

3x < 6 |

x < 2 |

(This means that when the value of x is less than 2, the inequality 4x - 5 < x + 1 is true.)

The above example is a **linear inequality** (there are no powers of x other than x^{1}, and it was solved as though it were a normal equation).

All inequalities are the same, except there are one or two tricks to spot. Principally when working with negatives, or taking reciprocals.

**Please note that for the following examples the inequality sign has changed.** (Can you see why?)

**Example**

The same is true for taking reciprocals. In this case we multiply and divide the quantities as follows:

**Example**

Check this yourself by multiplying both sides by 5x and dividing both sides by 12. (An extra check is to substitute in values and see if the solution works.)

As when solving quadratic equations preferably factorise, (or use the formula), and then sketch the curve remembering that a + x^{2} is a ∪-shape and a - x^{2} is a ∩-shape.

**Example:**

Solve 3x^{2} < 7x + 6

3x^{2} - 7x - 6 < 0

(3x + 2)(x - 3) < 0

The graph equals zero when x = -2/3 or x = 3

As the graph is a ∪-shape, the graph is below the y-axis when -2/3 < x < 3

Once a function is factorised it is possible to find where it equals zero, and by considering numbers in between these values we can determine where the curve is positive and negative and hence solve the inequality.

**For example:**

Solve (x-1)(x+2)(x+4) ≥ 0

It equals zero when x = 1, -2 and -4.

To find out what occurs when x takes other values we complete a quick table.

Range of x: | Sign of each bracket: | Overall sign: |
---|---|---|

x < -4 | (-)(-)(-) | -ve |

x = -4 | 0 | |

-4 < x < -2 | (-)(-)(+) | +ve |

x = -2 | 0 | |

-2 < x < 1 | (-)(+)(+) | -ve |

x = 1 | 0 | |

x > 1 | (+)(+)(+) | +ve |

A quick sketch of the curve will show these results more clearly:

So, the solutions are -4 ≤ x ≤ 2 and x ≥ 1.

* Note:* If you know the shape of the graph the table is no longer necessary as you can fill in the details once you've plotted the points where the graph crosses the x- and y-axes.