|3-r|<5
If r is >= 8, then it is not true. (|3-8| = 5)
If r is <= -2, then it is not true, (|3-(-2)| = 5)
So r is a member of the interval ]-2, 8[, carefully not including either -2 or 8 themselves.
How do you solve multi-step inequalities with fractions?
To solve the inequality |3 - r| < 5, you need to isolate the variable r.
1. Start by splitting the inequality into two separate cases: one for when the expression inside the absolute value is positive and one for when it's negative.
Case 1: 3 - r > 0 (when the expression inside the absolute value is positive)
In this case, the absolute value will simply be equal to the expression itself. So we have:
3 - r < 5
Case 2: 3 - r < 0 (when the expression inside the absolute value is negative)
In this case, the absolute value will be equal to the opposite of the expression. So we have:
-(3 - r) < 5
2. Solve each case separately.
Case 1:
3 - r < 5
Subtract 3 from both sides:
-r < 5 - 3
-r < 2
Multiply both sides by -1 and flip the inequality sign since we're multiplying by a negative number:
r > -2
Case 2:
-(3 - r) < 5
Distribute the negative sign inside the parentheses:
-r + 3 < 5
Subtract 3 from both sides:
-r < 5 - 3
-r < 2
Multiply by -1 and flip the inequality sign:
r > -2
3. Combine the solutions from both cases.
The solution to the inequality |3 - r| < 5 is r > -2.