linear inequalities

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LINEAR INEQUALITIES

Hot upon the heels of our work on equations comes this inoffensive topic requiring little in the way of extra teaching or knowledge.

Consider the inequality 3x + 5 < 17 Which says:

“Which numbers, call them x, when multiplied by 3 and subsequently increased by 5 give an

answer less than 17 ?” Look difficult ? Just look at the solution:

We have 3x + 5 < 17 Subtract 5 from both sides 3x < 12 Divide both sides by 3 to get x < 4, i.e. “x can be any number less than 4.”

Simple!

Example 1. Solve the inequality 3(x − 5) > 10 − 2x.

Solution. “Greater than”Remove brackets to get 3x − 15 > 10 − 2x

Add 2x to both sides to get 5x − 15 > 10 Add 15 to both sides to get 5x > 25 Divide both sides by 5 to get x > 5. {The solution is thus given by any number greater than 5.}

We cannot help but notice the similarities with solving the equivalent equations!

Example 2. Solve the inequality 10 + 5(x + 3) ≤ 8x − 2. Solution. “Less than or equal to”Remove brackets to get 10 + 5x + 15 ≤ 8x − 2

Tidy up ! 25 + 5x ≤ 8x − 2. Subtract 5x from both sides to get 25 ≤ 3x − 2. Add 2 to both sides to get 27 ≤ 3xDivide both sides by 3 to get 9 ≤ x.

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NOTE.The solution to the previous example is given by 9 ≤ x which says

“9 is less than or equal to x.”This is the same as saying

“x is any number greater than or equal to 9.” Hence we see that the solution given by 9 ≤ x can be expressed as x ≥ 9 if we so prefer.

Our next example is slightly more difficult although the algebra content is nothing special.

Example 3.a) Rearrange the inequality 3 + 7n < 2n +22 into the form n < some number.

b) Given that n also satisfies the inequality 3n > 1, write down all the integer values of n that satisfy both inequalities.

Whole numbers

Solution.a) We have 3 + 7n < 2n + 22 Subtract 2n from both sides to get 3 + 5n < 22 Subtract 3 from both sides to get 5n < 19 Divide both sides by 5 to get n < 19

5 .

Now, 195 is the same as 38

10 which equals 3.8 and thus our solution is given by n < 3.8 which says that n is any number less than 3.8. b) 3n >1 means that n > 1

3 . So, combining this with our answer to a) means that we are

looking for all integers n which are greater than 13 but, at the same time less than 3.8.

The only possibilities are n = 1, n = 2 or n = 3

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Exercise 1. NO CALCULATORS.

1) a) Rearrange the inequality 5n − 4 > 3n + 11 into the form n > a number.

b) Write down the least whole number value of n which satisfies this inequality.

2) a) Rearrange the inequality 4n − 4 < 20 − 2n

into the form n < a number.

b) Write down the greatest integer value (whole number) of n which satisfies this inequality.

3) a) Rearrange the inequality 4n − 4 < 20 − n

into the form n < a number.

b) Write down the greatest whole number value of n which satisfies this inequality.

4) a) Rearrange the inequality 3 − 2n > 3n −24 into the form n < some number.


5) a) Rearrange the inequality 3 + 4n < 20 − n into the form n < some number.


6) a) Rearrange the inequality 3 − 2n > 3n −33 into the form n < some number.


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ANSWERS / SOLUTIONS.

1) a) n > 7.5, b) n = 8. 2) a) n < 4, b) n = 3. 3) a) n < 4.8, b) n = 4. 4) a) n < 5.4, b) n = 1, n = 2, n = 3, n = 4 or n = 5. 5) a) n < 3.4, b) n = 1, n = 2 or n = 3. 6) a) n < 7.2, b) n = 1, n = 2, n = 3, n = 4, n = 5, n = 6 or n = 7.

Solution to 4).a) We have 3 − 2n > 3n −24 Add 2n to both sides 3 > 5n − 24 Add 24 to both sides 27 > 5n

Divide both sides by 5 5.4 > n. {Since 27 545 10= etc.}

Which is the same as n < 5.4 b) We now have that 3n > 2 which means that n > 2

3 . Combining this with the answer to a) means that the only permissible integer values are n = 1, n = 2, n = 3, n = 4 or n = 5.

linear inequalities

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