Math Problem Statement

Which graph shows the solution set of the system of inequalities: x ≥ -2 and x - 2y - 2 < 0?

Solution

The question provides the following system of inequalities:

x \geq -2 \\ x - 2y - 2 < 0 \end{cases}$$ We need to determine which graph correctly represents the solution set for this system. ### Step 1: Analyzing Each Inequality 1. **First Inequality**: $$x \geq -2$$ - This inequality represents a vertical line at $$x = -2$$ with shading to the right (where $$x$$ values are greater than or equal to $$-2$$). 2. **Second Inequality**: $$x - 2y - 2 < 0$$ - Rearrange to solve for $$y$$: $$x - 2y - 2 < 0$$ $$-2y < -x + 2$$ $$y > \frac{1}{2}x - 1$$ - This inequality represents a line with slope $$\frac{1}{2}$$ and y-intercept $$-1$$, with shading above the line (since $$y$$ is greater than the line). ### Step 2: Identifying the Solution We are looking for a graph where: - There is a vertical boundary line at $$x = -2$$ with shading to the right. - There is an oblique boundary line with a slope of $$\frac{1}{2}$$ and y-intercept at $$-1$$, with shading above this line. ### Solution Upon examining the options in the image, **Graph C** correctly shows: - A shaded region to the right of $$x = -2$$. - A shaded region above the line $$y = \frac{1}{2}x - 1$$. Thus, the correct answer is **Graph C**. Would you like a deeper explanation, or do you have any questions? ### Related Questions 1. How would the solution set change if the inequality $$x \geq -2$$ was instead $$x > -2$$? 2. What would be the solution set if the inequality $$y > \frac{1}{2}x - 1$$ was changed to $$y \leq \frac{1}{2}x - 1$$? 3. How can you determine the slope and intercept of a line given in a general inequality form? 4. What is the difference between strict inequalities and inclusive inequalities in graphing? 5. How does the shading direction differ between $$y >$$ and $$y <$$ inequalities? ### Tip When graphing inequalities, always identify boundary lines and carefully determine shading based on whether the inequality is strict ($$<$$ or $$>$$) or inclusive ($$\leq$$ or $$\geq$$).

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Linear Inequalities
Graphing

Formulas

Vertical line inequality: x ≥ -2
Slope-intercept form: y = mx + b

Theorems

-

Suitable Grade Level

Grades 9-10

Related Recommendation

Solving a System of Linear Inequalities with x ≥ -2 and x - 2y - 2 < 0

Solution Set for Linear Inequalities System: x ≥ −2 and x − 2y − 2 < 0

Graphing a System of Inequalities: y > -2x + 3 and y ≥ 2x - 9

Graphing a System of Linear Inequalities: y > -2x + 3 and y ≥ 2x - 9

Graphing and Solving Systems of Inequalities: y < x - 2, y > -5x + 2