Graphs for Precalculus:: Inequalities and Regions Flashcards (30 Cards) — Study Online

Q: The graph shows the solution of an inequality in \(y\) only (a horizontal boundary, **dashed**). Write the inequality.

\(y < -2\). The boundary is the horizontal line \(y=-2\), dashed because \(y=-2\) itself is not included (strict \( < \)). The shading is below the line, where \(y\) is smaller than \(-2\).

Q: The graph shows the solution of an inequality: the shaded region and its **solid** boundary line, which crosses the axes at \((3,0)\) and \((0,2)\). Write the inequality in the form \(ax+by \le c\).

\(2x+3y \le 6\). The line through \((3,0)\) and \((0,2)\) has intercept form \(\dfrac x3+\dfrac y2=1\); multiply by 6: \(2x+3y=6\). It is solid (\(\le\)). The origin is in the shaded region and \(2(0)+3(0)=0 \le 6\) is true, so the shaded side is the side containing the origin.

Curriculum Overview

Learn inequalities by reading their graphs. 30 flashcards, each with a real shaded graph on the front: linear inequalities (dashed or solid boundary, test points), absolute value inequalities, systems of inequalities and their corner points, linear programming (maximum and minimum), and nonlinear regions (parabolas, circles, ellipses, rings and the areas of regions). Every answer is checked and explained. For Algebra 2, precalculus, SAT/ACT math and an introduction to calculus.

Topics & Key Concepts

inequalities graphing inequalities shading boundary line dashed solid test point systems of inequalities linear programming feasible region corner points absolute value inequality nonlinear inequalities area of a region algebra 2 precalculus SAT ACT

Sample Flashcard Questions & Answers

Showing 8 of 30 cards

Question #1 Active Recall

The graph shows the solution of an inequality: the shaded region and its **dashed** boundary line. Write the inequality.

Answer & Explanation:

\(y>2x-1\).

The boundary line passes through \((0,-1)\) and \((1,1)\), so its equation is \(y=2x-1\). It is **dashed** because the points on the line are not solutions (the inequality is strict, \(>\)). The shading is above the line, where \(y\) is larger. Check with the test point \((0,0)\): \(0>2(0)-1=-1\) is true, and \((0,0)\) is in the shaded region.

Question #2 Active Recall

The graph shows the solution of an inequality: the shaded region and its **solid** boundary line. Write the inequality.

Answer & Explanation:

\(y \le -x+3\).

The boundary line passes through \((0,3)\) and \((3,0)\), so it is \(y=-x+3\). It is **solid** because points on the line are solutions (\(\le\)). The shading is below the line, where \(y\) is smaller. Test point \((0,0)\): \(0 \le 3\) is true, and \((0,0)\) is shaded.

Question #3 Active Recall

The graph shows the solution of an inequality in \(x\) only (a vertical boundary, **solid**). Write the inequality.

Answer & Explanation:

\(x \ge 2\).

The boundary is the vertical line \(x=2\), solid because \(x=2\) is included. The shading is to the right of the line, where \(x\) is larger. Every point with \(x\) at least 2 is a solution, whatever its \(y\) is.

Question #4 Active Recall

The graph shows the solution of an inequality in \(y\) only (a horizontal boundary, **dashed**). Write the inequality.

Answer & Explanation:

\(y < -2\).

The boundary is the horizontal line \(y=-2\), dashed because \(y=-2\) itself is not included (strict \( < \)). The shading is below the line, where \(y\) is smaller than \(-2\).

Question #5 Active Recall

The graph shows the solution of an inequality: the shaded region and its **solid** boundary line, which crosses the axes at \((3,0)\) and \((0,2)\). Write the inequality in the form \(ax+by \le c\).

Answer & Explanation:

\(2x+3y \le 6\).

The line through \((3,0)\) and \((0,2)\) has intercept form \(\dfrac x3+\dfrac y2=1\); multiply by 6: \(2x+3y=6\). It is solid (\(\le\)). The origin is in the shaded region and \(2(0)+3(0)=0 \le 6\) is true, so the shaded side is the side containing the origin.

Question #6 Active Recall

The graph shows the solution of an inequality: the shaded region and its **solid** boundary line. Write the inequality.

Answer & Explanation:

\(y \ge \dfrac{1}{2}x+1\).

The line has y-intercept \((0,1)\) and slope \(\dfrac12\) (it passes through \((2,2)\)), so it is \(y=\dfrac12x+1\). It is solid (\(\ge\)). The shading is above the line (larger \(y\)). Test point \((0,3)\): \(3 \ge 1\) is true and it is shaded.

Question #7 Active Recall

The graph shows the solution of \(y < 3x-1\) (shaded below a **dashed** line). Which of the points \((2,0)\), \((1,2)\) and \((0,0)\) are solutions of the inequality?

Answer & Explanation:

Only \((2,0)\) is a solution.

\((2,0)\): \(0 < 3(2)-1=5\) is true, and the point is in the shaded region. \((1,2)\): \(2 < 3(1)-1=2\) is false (the point is exactly **on the dashed line**, and the line is not included). \((0,0)\): \(0 < 3(0)-1=-1\) is false (the point is above the line, not in the shading).

Question #8 Active Recall

The graph shows the solution of an inequality in \(x\): a vertical strip between two **dashed** lines. Write it with an absolute value, and as a compound inequality.

Answer & Explanation:

\(|x| < 3\), that is \(-3 < x < 3\).

The boundary lines are \(x=-3\) and \(x=3\), dashed because the endpoints are not included. The solutions are the points whose distance from the y-axis is less than 3. In interval notation, \(x\) is in \((-3,3)\).

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