Solve -3(x-10)>-x+5. Write your solution using set builder notation. please show work
To solve the inequality -3(x-10) > -x + 5, you need to follow these steps:
Step 1: Distribute the -3 to both terms inside the parentheses:
-3(x) - 3(-10) > -x + 5
-3x + 30 > -x + 5
Step 2: Simplify both sides of the inequality:
-3x + 30 > -x + 5
Step 3: Get rid of the parentheses by removing unnecessary parentheses:
-3x + 30 > -x + 5
Step 4: Combine like terms on both sides:
-2x + 30 > 5
Step 5: Subtract 30 from both sides of the inequality to isolate the variable on the left side:
-2x + 30 - 30 > 5 - 30
-2x > -25
Step 6: Divide both sides by -2, remembering that dividing by a negative number reverses the inequality sign:
(-2x)/(-2) < (-25)/(-2)
x < 25/2
So, the solution to the inequality is x < 25/2.
Now, let's express the solution using set-builder notation.
The set-builder notation for the solution x < 25/2 can be written as:
{x | x is a real number, and x is less than 25/2}
or
{x ∈ ℝ | x < 25/2}