Use a graphing calculator to solve each inequality. Write the solution set in interval notation. See Using Your Calculator: Solving Inequalities Graphically
The Correct Answer and Explanation is:
To solve inequalities graphically using a graphing calculator, follow these steps:
1. Enter the Inequality Function:
- You need to input the inequality as a function. For example, if your inequality is x2−4x+3≥0x^2 – 4x + 3 \geq 0x2−4x+3≥0, you would first enter f(x)=x2−4x+3f(x) = x^2 – 4x + 3f(x)=x2−4x+3 into the graphing calculator.
2. Graph the Function:
- After entering the function, press the “Graph” button. The graph will display the function on the coordinate plane.
3. Analyze the Graph:
- Look for the points where the function is greater than or equal to 0 (if solving a “greater than or equal to” inequality), or less than or equal to 0 (if solving a “less than or equal to” inequality).
- For a quadratic like x2−4x+3x^2 – 4x + 3×2−4x+3, you’d look for where the graph intersects the x-axis, as the x-values where the graph is above or below the x-axis correspond to the solution set.
4. Use the Calculator’s “Trace” or “Intersection” Feature:
- Most graphing calculators have a feature that allows you to trace along the graph or find intersections. This helps you locate where the function crosses the x-axis.
- If solving f(x)≥0f(x) \geq 0f(x)≥0, locate the x-values where the graph is above the x-axis (or on it, if it touches the x-axis).
- If solving f(x)≤0f(x) \leq 0f(x)≤0, locate where the graph is below the x-axis.
5. Find the Solution Set:
- Based on the graph and where the inequality holds true (whether above or below the x-axis), write the solution set in interval notation.
- For example, if the solution to x2−4x+3≥0x^2 – 4x + 3 \geq 0x2−4x+3≥0 is (−∞,1]∪[3,∞)(-\infty, 1] \cup [3, \infty)(−∞,1]∪[3,∞), the function is greater than or equal to 0 from x=−∞x = -\inftyx=−∞ to x=1x = 1x=1, and from x=3x = 3x=3 to ∞\infty∞.
Example:
If you have the inequality 2x+3<02x + 3 < 02x+3<0, you would:
- Graph y=2x+3y = 2x + 3y=2x+3.
- Look for where the graph is less than 0 (where it’s below the x-axis).
- Find the x-value of the intersection point, which is x=−32x = -\frac{3}{2}x=−23.
- The solution in interval notation is (−∞,−32)(-\infty, -\frac{3}{2})(−∞,−23).
In summary, the graphing calculator helps you visualize the function’s behavior and find where the inequality holds true. After graphing, you can translate that into the solution set in interval notation.
