Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. \(3-5 x \leq-7\)

Short answer

[2, ∞)

Step-by-step solution

- Isolate the variable term

Subtract 3 from both sides of the inequality: \(3-5x - 3 \leq -7 - 3\)

- Simplify the inequality

Combine like terms: \( -5x \leq -10\)

- Solve for the variable

Divide both sides by -5 and remember to reverse the inequality sign (because we are dividing by a negative number): \( x \geq 2\)

- Express the solution in interval notation

Since \(x\) is greater than or equal to 2, the interval notation is: \([2, \infty)\)

- Graph the solution

Plot a number line, and shade the region starting from 2 and extending to infinity. Use a closed circle at 2 to indicate that 2 is included in the solution set.

Key concepts

Isolate the Variable

To solve an inequality, the first step is to isolate the variable (in this case, x) on one side of the inequality sign. In our problem, the inequality is:
\(3 - 5x \leq -7\).
By isolating the variable, we mean getting x by itself. This involves several mathematical operations. Here's how you can do this:
1. Start by removing any constants on the same side as the variable. Subtract 3 from both sides of the inequality:
\(3 - 5x - 3 \leq -7 - 3\).
2. Simplify the equation by combining like terms, which gives us:
\(-5x \leq -10\).
3. Now, divide both sides by -5. Remember to reverse the inequality sign because you are dividing by a negative number:
\(x \geq 2\).
After these steps, the variable x is isolated and we have solved for it.

Interval Notation

Once you have isolated the variable and found the solution, the next step is to express the solution in interval notation. This format helps in representing the set of all possible values that satisfy the inequality.
From the previous step, we learned that \(x \geq 2\).
In interval notation, this is represented as \([2, \infty)\). Here's a quick guide on interval notation:
* Square brackets \([]\) mean that the endpoint is included in the interval. Example: \([2, \infty)\) means 2 is included.
* Parentheses \(()\) mean that the endpoint is not included in the interval. Example: \((2, \infty)\) means 2 is not included.
* \(\infty\) and \(-\infty\) always take parentheses because they are not actual numbers and can't be 'included.'
So, remembering these rules can help you express any solution set accurately and clearly in interval notation.

Graphing Inequalities

Graphing inequalities involves plotting the solution set on a number line. This visual representation makes it easier to understand which values are included. Here’s how you can graph the solution \(x \geq 2\):
1. Draw a horizontal number line.
2. Locate the point 2 on the number line.
3. Since the inequality includes 2 (\(x \geq 2\)), draw a closed circle (or dot) at point 2. A closed circle signifies that the point is included in the solution set.
4. Shade the number line to the right of 2, extending to infinity, as these represent all values greater than or equal to 2.
By following these steps, you create a clear and accurate graph of the solution set, making it easy to visualize the range of values that satisfy the inequality.