LINEAR INEQUALITIES 2

                LINEAR INEQUALITIES

You will remember that a number line is a line running from left to right (horizontal) having numbers shown on it as shown below:

-5            -4            -3            -2             -1             0                1             2             3             4               5           x

Example 1 : Represent the inequality x > 3 on the number line

-3            -2            -1             0               1             2                3             4             5             6               7           x 

X > 3 means all numbers greater than 3 I e 4,5,6,7 etc

3 is excluded from these numbers to show this, we have an empty circle around the number 3

Example 2 : Represent x ≥ 3 on the number line

-3            -2            -1             0               1             2                3             4             5             6               7           x 

X > 3 means all numbers greater than 3.  I e 4,5,6,7 etc

3 is included in these numbers to show this, we have a shaded circle around the number 3.

Example 3: Solve the inequality x + 4 < 6

Solution

  X +4 < 6

Subtract 4 from both sides

                             X + 4 – 4< 6 – 4

X < 2

Example 4: Solve the inequality 6 ≤ 2x – 1 and sketch the graph of the solution.

 Solution

 Add 1 to both sides

6 + 1 ≤ 2x – 1 + 1

7 ≤ 2x

Divide both sides by 2
  ≤

3≤ x, then x ≤ 3

-3            -2            -1             0               1             2           3     3        4             5             6               7           x 

Multiplication and Division by negative numbers

As a rule, if both sides of an inequality are multiplied or divided by a negative number, the inequality must be reversed.        I.e.>   ≥  become   <   ≤.

Example 5:Solve 19≥ 4 – 5x

Solution

19 ≥ 4 – 5x

                Subtract 4 from both sides

19 – 4 ≥ -5x

15 ≥  -5x

Divide both sides by -5 and reverse the inequality sign

 ≤

-3 ≤ x

Example 6: Solve 5 – x > 3

Solution

          Subtract 5 from both sides

-x > -2

Multiply both sides by -1 and reverse the inequality

(-1) x (-x) < (-1) x(-2)

  X <2

Word problems involving inequalities

Example 7: if 9 is added to a number x the result is greater  than  17. Find x?

Solution

Let the number be x.

x + 9 > 17

Subtract 9 from both sides

x + 9 – 9 > 17 – 9

         X> 8

Example 8:  A triangle has sides of xcm, (x+7) cm and 10cm, where x is a whole number. If the perimeter of the triangle is less than 31, find the possible value of x.

Solution

                                                Perimeter  of a triangle ={ x + (x+7) +10}cm

     Thus, { x + x+7 +10} < 31x+x=2

  2x + 17 <31

Subtract 17 from both sides

2x + 17 – 17 = 31 – 17

  2x < 14

                                                                                Divide both sides by 2

<

X < 7

Also , in any triangle, the sum of the length of any two sides must be greater than the length of the third sides.  

                                                                                  Thus, x + (x+7) > 10

                                                                                2x + 7 > 10

                                                                                2x > 10 – 7

                                                                                2x > 3

X > 1

Thus x < 7and x > 1. But x must be a whole number. Thus the possible values of x are 2, 3 4, 5, 6.

Assignment

• Solve the following inequalities and represent their solutions on the number line.
• 3x  –  6  <  6
• 5x  +  5  ≥  30
• The circumference ( c ) of a circle is not less than 77cm. Find the range of the values of the radius. (Take π =  ).