The inequality
is shown above.
Which of the following inequalities is equivalent to the inequality above?
For inequality equivalence questions, first look for a greatest common factor you can divide from all terms; perform that operation on both sides, being very careful about whether the number is positive or negative (flip the sign only for negative). Then compare your simplified inequality to the answer choices, focusing on both the inequality direction and the coefficients/constant. This approach is quick, avoids unnecessary graphing, and minimizes sign errors.
Hints
Think about how to simplify the inequality
Look at the coefficients in . Do they share a common factor that you could divide out from every term?
Use the greatest common factor
Factor out of the left-hand side of . Rewrite the inequality using that factorization before simplifying further.
Be careful with the inequality sign
When you divide both sides of an inequality by a positive number, does the direction of the inequality sign change or stay the same?
Desmos Guide
Graph the original inequality
In Desmos, type 3x + 9y >= -12. You should see a boundary line and a shaded half-plane showing all the solutions to the original inequality.
Graph each answer choice one by one
On new lines in Desmos, enter each option exactly:
x + 3y <= -43x + 9y <= -12-x - 3y >= 4x + 3y >= -4
Turn them on and off (using the checkboxes) to compare each graph with the original.
Compare the shaded regions
Look for the choice whose boundary line and shaded region perfectly overlap with the graph of 3x + 9y >= -12—same line and same side shaded. That option is the inequality that is truly equivalent to the original.
Step-by-step Explanation
Understand what “equivalent inequality” means
Two inequalities are equivalent if they have exactly the same set of solutions. You can create an equivalent inequality by doing the same operation to both sides (like adding, subtracting, multiplying, or dividing by the same nonzero number), remembering that multiplying or dividing by a negative number flips the inequality sign.
Look for a common factor to simplify the inequality
Start with the given inequality:
Notice that every term (, , and ) is divisible by . Factor out from the left side:
So the inequality can be rewritten as:
Divide both sides by 3 and match the result to a choice
Now divide both sides of the inequality by to isolate the parentheses:
Because is positive, the inequality sign stays the same:
This simplified inequality is equivalent to the original one and matches answer choice D, .