/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 60 Solve the inequality \(3 x+2>... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 60

Solve the inequality \(3 x+2>14,\) and graph the solution set on a number line.

Short Answer

Expert verified
The solution is \( x > 4 \).

Step by step solution

01

- Isolate the variable

Start by isolating the variable on one side of the inequality. Subtract 2 from both sides: \[ 3x + 2 - 2 > 14 - 2 \] Simplifying this, we get: \[ 3x > 12 \]
02

- Solve for the variable

Next, solve for the variable by dividing both sides of the inequality by 3: \[ \frac{3x}{3} > \frac{12}{3} \] Simplifying this, we get: \[ x > 4 \]
03

- Graph the solution set

To graph the solution on a number line, draw a line and mark the point 4. Since the inequality is strict (greater than but not equal to), use an open circle at 4. Shade the line to the right of 4 to indicate all numbers greater than 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the variable
To solve an inequality, our first goal is to isolate the variable on one side. This step sets the stage for solving the problem completely. Let's take a closer look at the example inequality:
In the problem, we start with: 3x + 2 > 14 The objective is to get 'x' alone on one side of the inequality. We achieve this by performing the same operation on both sides of the inequality. Here, we subtract 2 from both sides, which gives us: 3x > 12 Once we've isolated the variable, we can move on to the next step. Remember, subtracting the same amount from both sides doesn't change the direction of the inequality sign!
Inequality sign
The inequality sign is crucial and dictates the direction of our solution set. Here, we started with:
3x + 2 > 14 The '>' symbol means 'greater than', so the solution will involve values greater than the final number. We then proceeded to isolate the variable and ended up with: x > 4 This tells us that 'x' can be any number greater than 4. It's important to handle the inequality sign carefully, especially when multiplying or dividing by negative numbers, as it can change direction. However, we only divided by a positive number (3), so the sign doesn't change in this case.
Graphing solution set
Finally, let's graph the solution set. Graphing an inequality helps visually represent the range of possible solutions. In our solved inequality, we have:
x > 4 To represent this on a number line:
  • Draw a number line and mark the point 4.
  • Since 'x' is greater than 4 but not equal to 4, use an open circle at 4.

  • Shade the region to the right of 4 to indicate all numbers greater than 4.
This visually shows that any number to the right of 4 on the number line is a valid solution to the inequality. Remember, the open circle signifies that 4 is not included in the solution set.

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