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Exercises Graphs and straight lines

Example 4.70 from Elementary Algebra, 4.7 Graphs of Linear Inequalities

Graphs of Linear Inequalities: example 4.70

The boundary line shown is $y = 2 x - 1$ . Write the inequality shown by the graph.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2 x minus 1 is plotted as a solid arrow extending from the bottom left toward the upper right. The coordinate plane to the left of the line is shaded

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The line $y = 2 x - 1$ is the boundary line. On one side of the line are the points with $y > 2 x - 1$ and on the other side of the line are the points with $y < 2 x - 1$ .

Let’s test the point $(0, 0)$ and see which inequality describes its side of the boundary line.

At $(0, 0)$ , which inequality is true:

$\begin{matrix} y > 2 x - 1 & or & y < 2 x - 1 ? \\ y > 2 x - 1 & y < 2 x - 1 \\ 0 \overset{?}{>} 2 \cdot 0 - 1 & 0 \overset{?}{<} 2 \cdot 0 - 1 \\ 0 > −1 True & 0 < −1 False \end{matrix}$

Since, $y > 2 x - 1$ is true, the side of the line with $(0, 0)$ , is the solution. The shaded region shows the solution of the inequality $y > 2 x - 1$ .

Since the boundary line is graphed with a solid line, the inequality includes the equal sign.

The graph shows the inequality $y \geq 2 x - 1$ .

We could use any point as a test point, provided it is not on the line. Why did we choose $(0, 0)$ ? Because it’s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that $y < 2 x - 1$ .

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Write the inequality shown by the graph with the boundary line $y = −2 x + 3$ .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 2 x plus 3 is plotted as a solid arrow extending from the top left toward the bottom right. The coordinate plane to the right of the line is shaded.

How did it go?

PreviousGraphs of Linear Inequalities: example 4.69 Next Graphs of Linear Inequalities: example 4.71

From Elementary Algebra (1st edition) by OpenStax (Rice University), section 4.7, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.