Solving a Conjunction
Solve conjunctions (AND compound inequalities) in Grade 9 algebra by isolating the variable in each inequality and finding the intersection of both solution sets, written as a combined interval.
Key Concepts
Property A conjunction can be solved as two inequalities connected with AND or as a single inequality with three parts, such as $30 \le 10 + 0.10x \le 40$. Explanation Solving a three part inequality is like a balancing act! To find your variable, you must perform the same operation on all three parts—the left, the middle, and the right. This ensures that you keep the relationship between the numbers fair and balanced while you work to isolate the variable in the center. Examples A phone bill between 30 and 40 dollars, with a 10 dollars fee and 0.10 dollars per minute, $x$, is $30 \le 10 + 0.10x \le 40$. First, subtract 10 from all three parts: $20 \le 0.10x \le 30$. Next, divide all three parts by 0.10 to find the range of minutes: $200 \le x \le 300$.
Common Questions
What is a conjunction in algebra?
A conjunction is a compound inequality joined by AND, requiring both conditions to be true simultaneously. It takes the form a < x < b, meaning x must be greater than a AND less than b at the same time.
What steps solve a conjunction like -3 < 2x + 1 < 9?
Treat the conjunction as two inequalities and apply the same operations to all three parts. Subtract 1 throughout: -4 < 2x < 8. Divide by 2: -2 < x < 4. The solution is the interval (-2, 4).
How do you graph the solution of a conjunction on a number line?
Mark both boundary values on the number line with open circles (for strict inequalities) or closed circles (for ≤ or ≥), then shade the region between them. The shaded segment shows all values satisfying both conditions.