How many integer values of x are there such that |x−3|+5<16?
To find the number of integer values of x that satisfy the inequality |x−3|+5<16, we need to solve the inequality and count the number of integer solutions.
Step 1: Subtract 5 from both sides of the inequality:
|x−3|<11.
Step 2: Break down the absolute value into two separate cases:
Case 1: x−3<11.
Case 2: x−3>(−11) ⇒ x−3>−11.
Step 3: Solve each case separately:
Case 1: x−3<11 ⇒ x<14.
Case 2: x−3>−11 ⇒ x>−8.
Step 4: Combine the solutions from both cases:
The solution to the inequality |x−3|<11 is given by x<14 and x>−8.
Step 5: Count the number of integer solutions within the given range:
Since x has to be an integer, we need to find the number of integers in the range from −8 to 13 (inclusive), as this includes all possible values of x that satisfy the inequality.
The number of integers within this range can be calculated by subtracting the lower boundary from the upper boundary and adding 1:
Number of integers = Upper boundary - Lower boundary + 1
= 13 - (-8) + 1
= 21.
Therefore, there are 21 integer values of x that satisfy the inequality |x−3|+5<16.