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Symbols, solving inequalities and graphs

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# Solving inequalities

An inequality such as 3x - 7 < 8 is similar to the equation 3x - 7 = 8. However, this time we are looking for numbers which if you multiply by 3, then subtract 7, you get an answer of less than 8.

Unlike 3x - 7 = 8, which has just one answer, there are lots of numbers for which this is true (in fact, an infinite number). So our answer is not a number, but a range of numbers.

We solve these inequalities just like simple equations: what you do to one side, you must do to the other.

Now have a look at this typical question:

Question

Solve the expression 3x - 7 < 8.

• First, write down the inequality:
• 3x - 7 < 8
• Then add 7 to both sides, to cancel out the -7:
• 3x < 15
• Next, simplify the inequality by dividing both sides by the number in front of x - in this case 3.
• x < 5

So the inequality in 3x - 7 < 8 is satisfied when x is less than 5. (Note that this does not include 5 itself.)

Sometimes the unknown appears on both sides, just as in simple equations.

Now look at this question.

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