Inequality in Two Variables

Examples;

1. State the range of values of x for the graph below;

Solution

               x < -1               or               x ≥ +4

Therefore, the range of values of x will be;

-1 > x ≥ +4.

2. State the range of values of y for the graph below;

        

Solution

                        x ≤ -5            or           x ≥ 9

Therefore, the range of values of x will be;

-5 ≥ x ≥ 9.

3. If 3 + x ≤ 5 and 8 + x ≥ 5, what range of values of x satisfies both inequalities?

Solution

3 + x ≤ 5        or       8 + x ≥ 5

(Making x the subject)

 x ≤ 5 – 3        or      x ≥ 5 – 8

   x ≤ 2           or      x ≥ - 3

Therefore, the range will be;

2 ≥ x ≥ -3.

Practice these questions below;

1. State the range of values of x for the graph below;

2. If 6x < 2 – 3x and x – 7 < 3x, what range of values of x satisfies both inequalities?

3. What is the range of values of x for which 3(1 – x) > 3 and 3(1 + x) ≥ 0 are both satisfied?

4. State the range of values of x for the graph below;

5. What is the range of values of y for which 4y – 7 ≤ 3y and 3y ≤ 5y + 8 are both satisfied?

Linear Inequalities

An inequality is just like linear equation which has two sides and an unknown variable which are always represented with alphabets (a-z) but deals with a lot of signs.

Signs of inequalities

> Greater than

< Less than

≥ Greater or equal to

≤ Less than or equal to

Solve the following inequalities;

Example 1; Simplify 2x – 4 > 8

Solution

Note;

In order to do this, you must consider the two sides of this inequality.

2x – 4 > 8 (Make x the subject of the equation)

2x > 8 + 4

2x > 12 (Divide both sides by 2)

x > 6.

Example 2; Solve the inequalities 3x – 8 ≤ 10 + 5x

Solution

3x – 8 ≤ 10 + 5x (Collect like terms)

3x – 5x ≤ 10 + 8

- 2x ≤ 18 (Divide both sides by the co-efficient of x which is -2)

x ≥ - 9.

Note; the inequality sign must change when dividing with a minus sign).

Practice these questions below;

1. Simplify 4x – ¹/₃   ≥   ²/_3x + 2

3. Simplify 6x – 2 < 2x + 8

4. Simplify ¹/_3y  >  ¹/_2y  +  ¹/₄

5. Simplify ⁶/_x  ≥  2