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Equations and inequalities: linear inequalities

Investigation : Inequalities on a Number Line

Represent the following on number lines:

  1. x = 4 x = 4
  2. x < 4 x < 4
  3. x ≤ 4 x ≤ 4
  4. x ≥ 4 x ≥ 4
  5. x > 4 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 2 x + 2 ≤ 1 2 - x 3 x + 1 ≥ 2 4 3 x - 6 < 7 x + 2
(1)

The methods used to solve linear inequalities are identical to those used to solve linear equations. The only difference occurs when there is a multiplication or a division that involves a minus sign. For example, we know that 8>68>6. If both sides of the inequality are divided by -2-2, -4-4 is not greater than -3-3. Therefore, the inequality must switch around, making -4<-3-4<-3.

Tip:

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<1x<1, then -x>-1-x>-1.

In order to compare an inequality to a normal equation, we shall solve an equation first. Solve 2x+2=12x+2=1.

2 x + 2 = 1 2 x = 1 - 2 2 x = - 1 x = - 1 2 2 x + 2 = 1 2 x = 1 - 2 2 x = - 1 x = - 1 2
(2)

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

Figure 1
Figure 1 (MG10C10_001.png)

Now let us solve the inequality 2x+2≤12x+2≤1.

2 x + 2 ≤ 1 2 x ≤ 1 - 2 2 x ≤ - 1 x ≤ - 1 2 2 x + 2 ≤ 1 2 x ≤ 1 - 2 2 x ≤ - 1 x ≤ - 1 2
(3)

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

Figure 2
Figure 2 (MG10C10_002.png)

As you can see, for the equation, there is only a single value of xx 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.

Figure 3
Khan academy video on inequalities - 1

Figure 4
Khan academy video on inequalities - 2

Exercise 1: Linear Inequalities

Solve for rr: 6-r>26-r>2

Solution

  1. Step 1. Move all constants to the RHS :
    - r > 2 - 6 - r > - 4 - r > 2 - 6 - r > - 4
    (4)
  2. Step 2. Multiply both sides by -1 :

    When you multiply by a minus sign, the direction of the inequality changes.

    r < 4 r < 4
    (5)
  3. Step 3. Represent answer graphically :

    Figure 5
    Figure 5 (MG10C10_003.png)

Exercise 2: Linear Inequalities

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

Solution

  1. Step 1. Expand all brackets :
    4 q + 3 < 2 ( q + 3 ) 4 q + 3 < 2 q + 6 4 q + 3 < 2 ( q + 3 ) 4 q + 3 < 2 q + 6
    (6)
  2. Step 2. Move all constants to the RHS and all unknowns to the LHS :
    4 q + 3 < 2 q + 6 4 q - 2 q < 6 - 3 2 q < 3 4 q + 3 < 2 q + 6 4 q - 2 q < 6 - 3 2 q < 3
    (7)
  3. Step 3. Solve inequality :
    2 q < 3 Divide both sides by 2 q < 3 2 2 q < 3 Divide both sides by 2 q < 3 2
    (8)
  4. Step 4. Represent answer graphically :

    Figure 6
    Figure 6 (MG10C10_004.png)

Exercise 3: Compound Linear Inequalities

Solve for xx: 5≤x+3<85≤x+3<8 and represent solution on a number line.

Solution

  1. Step 1. Subtract 3 from Left, middle and right of inequalities :
    5 - 3 ≤ x + 3 - 3 < 8 - 3 2 ≤ x < 5 5 - 3 ≤ x + 3 - 3 < 8 - 3 2 ≤ x < 5
    (9)
  2. Step 2. Represent answer graphically :

    Figure 7
    Figure 7 (MG10C10_005.png)

Linear Inequalities

  1. Solve for xx and represent the solution graphically:
    1. 3x+4>5x+83x+4>5x+8
    2. 3(x-1)-2≤6x+43(x-1)-2≤6x+4
    3. x-73>2x-32x-73>2x-32
    4. -4(x-1)<x+2-4(x-1)<x+2
    5. 12x+13(x-1)≥56x-1312x+13(x-1)≥56x-13
    Click here for the solution
  2. Solve the following inequalities. Illustrate your answer on a number line if xx is a real number.
    1. -2≤x-1<3-2≤x-1<3
    2. -5<2x-3≤7-5<2x-3≤7
    Click here for the solution
  3. Solve for xx: 7(3x+2)-5(2x-3)>77(3x+2)-5(2x-3)>7
    Illustrate this answer on a number line.
    Click here for the solution

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