Solve and graph inequalities

Learning to solve and graph inequalities is easy and straightforward once you know how to solve two-step equations.

Example #1:

Solve x + 3 < 9

x + 3 < 9

Subtract 3 from each side of the equation

x + 3 - 3 < 9 - 3

x + 0 < 6

x < 6

Graph the solution

x is smaller than 6

Check:

Choose a point on the graph such as 4.

4 + 3 = 7 and 7 is smaller than 9

Example #2:

Solve 5x - 2 ≥ 13

5x - 2 ≥ 13

Add 2 to each side of the equation

5x - 2 + 2 ≥ 13 + 2

5x + 0 ≥ 15

5x  ≥ 15

Divide each side by 5

5x/5  ≥ 15/5

x  ≥ 3

Graph the solution

x is bigger than or equal to 3

Notice how this time the circle is shaded at 3 meaning that 3 is included in the solution.

Check:

Choose a point on the graph such as 7.

5(7) - 2 = 35 - 2 = 33 and 33 is bigger than 7.

Example #3:

Solve 2 + 3(5 - x)  ≥ 38

2 + 3(5 - x)  ≥ 38

Use the distributive property 

2 + 15 - 3x  ≥ 38

Simplify by adding 2 and 15

17 - 3x  ≥ 38

Subtract 17 from each side of the equation

17 - 17 - 3x  ≥ 38 - 17

0 - 3x  ≥ 21

-3x  ≥ 21

Divide each side by -3 and reverse the inequality

-3x/-3  ≤ 21/-3

x ≤ -7

Graph the solution

x is smaller than or equal to negative seven

Check:

Choose a point on the graph such as -10.

2 + 3(5 - -10) = 2 + 3(5 + +10) = 2 + 3(15) = 2 + 45 = 47 and 47 is bigger than 38.

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