x-8 is less than or equal to 4 or 2x+3 is greater than 9

Question

How do you solve this inequality that is here below?

x-8 is less than or equal to 4 or 2x+3 is greater than 9 

Answer 1

When these inequalities are joined by the word "or", it means that we can use the solution from the first inequality OR the second inequality.  On a number line we would keep both solutions.

 

x-8<4   OR   2x+3 > 9

 

Solve each one separately.

 

x - 8 < 4

 

Add 8 to both sides.

 

x < 12                OR

 

 

 

2x+3>9

 

Subtract 3 from both sides

 

2x>6

 

Divide both sides by 2

 

x > 3

 

 

 

So your answer consists of two parts,  x < 12   or  x > 3

 

If you would graph this on the number line, you would see that the whole number line is included in the answer.

 

So x is all real numbers.

 

Check:  If x = 8, it is both less than 12 and greater than 3

 

If x = -100, it is less than 12, so it satisfies the first inequality

 

If x = +700, it is greater than 3, so it satisfies the second inequlity.

 

Any real number will satisfy one or both of the inequalities.  So the answer is ALL REAL NUMBERS.