In Exercises 63–64, write each sentence as an inequality in two variables. Then graph the inequality. The y-variable is at least 4 more than the product of -2 and the x-variable.

Verified step by step guidance

1. Identify the variables: let $x$ be the x-variable and $y$ be the y-variable.
2. Translate the phrase 'the y-variable is at least 4 more than the product of -2 and the x-variable' into an inequality. 'At least' means 'greater than or equal to', and 'the product of -2 and the x-variable' is $-2x$. So the inequality is $y\; \backslash (\backslash geq\backslash )\; -2x\; +\; 4$.
3. Rewrite the inequality in slope-intercept form if needed: $y\; \backslash (\backslash geq\backslash )\; -2x\; +\; 4$ is already in slope-intercept form, where the slope is $-2$ and the y-intercept is $4$.
4. To graph the inequality, first graph the boundary line $y\; =\; -2x\; +\; 4$. Since the inequality is 'greater than or equal to', the boundary line should be solid.
5. Determine which side of the line to shade by testing a point not on the line, such as $(0,0)$. Substitute into the inequality: $0\; \backslash (\backslash geq\backslash )\; -2(0)\; +\; 4$ becomes $0\; \backslash (\backslash geq\backslash )\; 4$, which is false. So shade the side of the line that does not include $(0,0)$.

Key Concepts

Translating Verbal Statements into Inequalities

This involves converting a sentence describing a relationship between variables into a mathematical inequality. Key words like 'at least' indicate 'greater than or equal to' (≥), and understanding how to express phrases such as 'the product of -2 and the x-variable' is essential for accurate translation.

Inequalities in Two Variables

An inequality in two variables, such as y ≥ expression involving x, represents a region on the coordinate plane. It defines all points (x, y) that satisfy the inequality, not just a line, and understanding this helps in graphing the solution set correctly.

Graphing Linear Inequalities

Graphing involves first drawing the boundary line from the related equation (using equality), then shading the region that satisfies the inequality. Knowing how to determine which side to shade based on the inequality symbol is crucial for visualizing the solution.