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Linear inequalities

Investigation : inequalities on a number line

Represent the following on number lines:

  1. x = 4
  2. x < 4
  3. x 4
  4. x 4
  5. x > 4

A linear inequality is similar to a linear equation in that the largest exponent of a variable is 1. The following are examples of linear inequalities.

2 x + 2 1 2 - x 3 x + 1 2 4 3 x - 6 < 7 x + 2

The methods used to solve linear inequalities are identical to those used to solve linear equations. The only difference occurs when there is amultiplication or a division that involves a minus sign. For example, we know that 8 > 6 . If both sides of the inequality are divided by - 2 , - 4 is not greater than - 3 . Therefore, the inequality must switch around, making - 4 < - 3 .

When you divide or multiply both sides of an inequality by any number with a minus sign, the direction of the inequality changes. For this reason you cannot divide or multiply by a variable.

For example, if x < 1 , then - x > - 1 .

In order to compare an inequality to a normal equation, we shall solve an equation first. Solve 2 x + 2 = 1 .

2 x + 2 = 1 2 x = 1 - 2 2 x = - 1 x = - 1 2

If we represent this answer on a number line, we get

Now let us solve the inequality 2 x + 2 1 .

2 x + 2 1 2 x 1 - 2 2 x - 1 x - 1 2

If we represent this answer on a number line, we get

As you can see, for the equation, there is only a single value of x for which the equation is true. However, for the inequality, there is a range of values for which the inequality is true. This is the main difference between an equation and an inequality.

Khan academy video on inequalities - 1

Khan academy video on inequalities - 2

Solve for r : 6 - r > 2

  1. - r > 2 - 6 - r > - 4
  2. When you multiply by a minus sign, the direction of the inequality changes.

    r < 4

Solve for q : 4 q + 3 < 2 ( q + 3 ) and represent the solution on a number line.

  1. 4 q + 3 < 2 ( q + 3 ) 4 q + 3 < 2 q + 6
  2. 4 q + 3 < 2 q + 6 4 q - 2 q < 6 - 3 2 q < 3
  3. 2 q < 3 Divide both sides by 2 q < 3 2

Solve for x : 5 x + 3 < 8 and represent solution on a number line.

  1. 5 - 3 x + 3 - 3 < 8 - 3 2 x < 5

Linear inequalities

  1. Solve for x and represent the solution graphically:
    1. 3 x + 4 > 5 x + 8
    2. 3 ( x - 1 ) - 2 6 x + 4
    3. x - 7 3 > 2 x - 3 2
    4. - 4 ( x - 1 ) < x + 2
    5. 1 2 x + 1 3 ( x - 1 ) 5 6 x - 1 3
  2. Solve the following inequalities. Illustrate your answer on a number line if x is a real number.
    1. - 2 x - 1 < 3
    2. - 5 < 2 x - 3 7
  3. Solve for x : 7 ( 3 x + 2 ) - 5 ( 2 x - 3 ) > 7 Illustrate this answer on a number line.

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Source:  OpenStax, Siyavula textbooks: grade 10 maths [ncs]. OpenStax CNX. Aug 05, 2011 Download for free at http://cnx.org/content/col11239/1.2
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