Use the drawing tools to form the correct answer on the number line. Graph the solution set to this inequality:
[tex]$-2 x+9 \leq 5 x-12$[/tex]
Answer (2)
Add 2 x to both sides: 9 ≤ 7 x − 12 .
Add 12 to both sides: 21 ≤ 7 x .
Divide both sides by 7 : 3 ≤ x .
The solution is x ≥ 3 , represented on a number line with a closed point at 3 and a ray extending to the right. x ≥ 3
Explanation
Understanding the Problem We are given the inequality − 2 x + 9 ≤ 5 x − 12 . Our goal is to solve for x and then represent the solution on a number line. This means we want to find all values of x that make the inequality true.
Adding 2 x to Both Sides First, let's isolate x on one side of the inequality. We can start by adding 2 x to both sides: − 2 x + 9 + 2 x ≤ 5 x − 12 + 2 x 9 ≤ 7 x − 12
Adding 12 to Both Sides Next, we add 12 to both sides of the inequality: 9 + 12 ≤ 7 x − 12 + 12 21 ≤ 7 x
Dividing by 7 Now, we divide both sides by 7 :
7 21 ≤ 7 7 x 3 ≤ x This is equivalent to x ≥ 3 .
Graphing the Solution The solution set is all x such that x is greater than or equal to 3 . On a number line, this is represented by a ray starting at 3 and extending to the right. Since x can be equal to 3 , we use a closed point (or a solid circle) at x = 3 to indicate that 3 is included in the solution.
Examples
Imagine you're saving money for a new video game that costs 5 x − 12 dollars. You currently have − 2 x + 9 dollars. The inequality − 2 x + 9 ≤ 5 x − 12 helps you determine the minimum value of x (perhaps representing weeks) you need to save so that you have enough money to buy the game. Solving this inequality tells you how many weeks you need to save to afford the game.
The solution to the inequality − 2 x + 9 ≤ 5 x − 12 is x ≥ 3 . This means that any value of x that is greater than or equal to 3 satisfies the inequality. On a number line, it is represented with a closed point at 3 and a ray extending to the right.
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