Why do you use the open point and not the closed point in a graphic inequalities?

Answer 1

A number line is used to draw the graph of the inequality. The graph of an inequality consists of the graphs of all its solutions.

If you graph x < 5 on a number line, the solution is all real numbers to the left of 5. Use an open dot at 5, to indicate that 5 is not a solution.

If you graph x > 5, the solution is all real numbers to the right of 5. Use an open dot at 5 to indicate that 5 is not a solution.

If you graph x <= 5, the solution is all real numbers to the left of 5, included 5. Use a closed dot at 5 to indicate that 5 is a solution.

If you graph x >= 5, the solution is all real numbers to the right of 5, included 5. Use a closed dot at 5 to indicate that 5 is a solution.

Inequalities that have the same solution set are called equivalent inequalities. Solving an inequality is a process of writing equivalent inequalities until you isolate the variable. To do this, you apply the addition, subtraction, multiplication, division and transitive properties.

Examples:

x + 9 <= 7 subtract 9 from each side;

x + 9 - 9 <= 7 - 9

x <= -2

All numbers less than or equal to -2 are solutions. (Use a closed dot at -2, to indicate that -2 is a solution).

-3(x - 5) > -6 use the distributive property;

-3x + 15 > -6 subtract 15 from each side;

-3x + 15 - 15 > -6 - 15

-3x > -21 divide each side of the inequality by -3, and reverse the order of the inequality (that is, change the symbol of the inequality) ;

x < 21/3

x < 7

All numbers less than 7 are solutions. (Use an open dot at 7, to indicate that 7 is not a solution).

Two inequalities joined by the words and or orform a compound inequality. For example,

x > -4 and x > 4 the word and signals a conjunction.

The solutions are all numbers that are solutions of both inequalities.

Use an open dot at -4 and another open dot at 4. All numbers greater than -4 and less than 4 are solutions.

x < -3 or x > 5 the word or signals a disjunction.

The solutions are all numbers that are solutions of either inequality.

Use an open dot at -3 and another open dot at 5. All numbers less than -3 and greater than 5 are solutions.

Examples:

-7 < 4 - x <= 1 write two inequalities joined by and, then solve each inequality;

-7 < 4 -x and 4 - x <= 1 subtract 4 from each side;

-7 - 4 < 4 - 4 - x and 4 - 4 -x <= 1 - 4

-11 < -x and -x <= -3 multiply each side by -1, and reverse the order;

11 > x and x >= 3 so, you have;

11 > x >= 3 that is,

3 <= x < 11

All numbers greater than or equal to 3 and less than 11 are solutions. Use a closed dot at 3 and an open dot at 11.

Or you can work like this:

-7 < 4 - x <= 1

-7 - 4 < 4 - 4 - x <= 1 - 4

-11 < - x <= -3

11 > x >= 3

3 <= x < 11

5x < -2 or x + 3 > 4

Solution:

5x < -2 or x + 3 > 4

5x/5 < -2/5 or x + 3 - 3 > 4 - 3

x < -0.4 or x > 1

All numbers less than -0.4 and greater than 1 are solutions. Use an open dot at -0.4 and another open dot at 1.