What is a correct first step in solving the following inequality?
$0-12+20 x \geq -6 x+9$
Answer (2)
Simplify each inequality by combining the constant terms.
Add 12 to both sides of any of the inequalities to further isolate the variable term.
For example, adding 12 to both sides of the third inequality results in 20 x ≥ − 6 x + 21 .
This manipulation is a correct first step in solving the inequalities.
Explanation
Understanding the Problem We are given three inequalities:
=" -6 x+9"> 0 − 12 − 20 x " >= " − 6 x + 9
0 − 12 + 20 x " <= " − 6 x + 9
=" -6 x+9"> 0 − 12 + 20 x " >= " − 6 x + 9
Our objective is to determine a correct first step in solving the given inequalities. A correct first step would be to simplify each inequality by combining constant terms.
Simplifying Inequality 1 For inequality 1: Combine 0 and -12 to get -12. The inequality becomes = -6x + 9"> − 12 − 20 x " >= − 6 x + 9
Simplifying Inequality 2 For inequality 2: Combine 0 and -12 to get -12. The inequality becomes − 12 + 20 x " <= " − 6 x + 9
Simplifying Inequality 3 For inequality 3: Combine 0 and -12 to get -12. The inequality becomes = -6x + 9"> − 12 + 20 x " >= − 6 x + 9
Adding 12 to Both Sides Now, we can choose any of the simplified inequalities and perform a valid algebraic manipulation. A common first step is to add 12 to both sides of each inequality to isolate the x terms and constant terms on opposite sides of the inequality. Let's demonstrate this with inequality 3:
Adding 12 to both sides of = -6x + 9"> − 12 + 20 x " >= − 6 x + 9 gives: = -6x + 9 + 12"> − 12 + 20 x + 12" >= − 6 x + 9 + 12 = -6x + 21"> 20 x " >= − 6 x + 21
Examples
Linear inequalities are used in various real-world scenarios, such as determining the minimum production level to achieve a certain profit, calculating the maximum number of items that can be purchased within a budget, or setting constraints in optimization problems. Understanding how to manipulate and solve these inequalities is crucial for making informed decisions in these situations.
The correct first step in solving the inequality 0 − 12 + 20 x ≥ − 6 x + 9 is to simplify the left side first, giving you − 12 + 20 x ≥ − 6 x + 9 . Then, you can add 12 to both sides to further isolate the variable terms. This leads to a simplified form of the inequality, making it easier to solve for x .
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