Solve the inequality.
[tex]2(4 x-3) \geq-3(3 x)+5 x[/tex]
Answer (2)
Expand the inequality: 2 ( 4 x − 3 ) g e − 3 ( 3 x ) + 5 x becomes 8 x − 6 g e − 9 x + 5 x .
Simplify the inequality: 8 x − 6 g e − 4 x .
Isolate x : 12 xg e 6 .
Solve for x : xg e 2 1 . Therefore, the solution is xg e 0.5 .
Explanation
Understanding the Inequality We are given the inequality 2 ( 4 x − 3 ) ≥ − 3 ( 3 x ) + 5 x and asked to solve for x . This involves expanding and simplifying the inequality to isolate x on one side.
Expanding the Inequality First, we expand both sides of the inequality: 2 ( 4 x − 3 ) ≥ − 3 ( 3 x ) + 5 x 8 x − 6 ≥ − 9 x + 5 x
Simplifying the Inequality Next, we simplify the right side of the inequality by combining like terms: 8 x − 6 ≥ − 4 x
Isolating x Now, we want to isolate x on one side. We can add 4 x to both sides of the inequality: 8 x − 6 + 4 x ≥ − 4 x + 4 x 12 x − 6 ≥ 0
Further Isolating x Then, we add 6 to both sides: 12 x − 6 + 6 ≥ 0 + 6 12 x ≥ 6
Solving for x Finally, we divide both sides by 12 to solve for x :
12 12 x ≥ 12 6 x ≥ 2 1 x ≥ 0.5
Expressing the Solution The solution to the inequality is x ≥ 0.5 . This means that x can be any value greater than or equal to 0.5. In interval notation, this is represented as [ 0.5 , ∞ ) .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, consider a budget constraint where you have a certain amount of money to spend. If you want to buy items that cost a certain amount each, you can use inequalities to determine the maximum number of items you can purchase without exceeding your budget. Similarly, in business, inequalities can help determine the minimum sales required to achieve a certain profit target. Inequalities are also used in science and engineering to model constraints and optimize designs.
To solve the inequality 2 ( 4 x − 3 ) ≥ − 3 ( 3 x ) + 5 x , we expand, simplify, and isolate x to find that the solution is x ≥ 0.5 . This means that x can take any value greater than or equal to 0.5 .
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