Find a linear inequality with the following solution set. Each grid line represents one unit.

(Give your answer in the form $ax+by>c$ or $ax+by\geq c$ where $a,$ $b,$ and $c$ are integers with no common factor other than $1$.)
[asy] size(175);
real xmin = -5.5, xmax = 5.5, ymin = -5.5, ymax = 5.5;
fill((xmin,16/3)--(xmin,ymax)--(xmax,ymax)--(xmax,-2)--cycle,yellow);
ticks tgrid = Ticks(Label("%"), Step = 1, extend = true, rgb(0,0.7,0.7) + 0.4*bp);
yaxis(LeftRight, ymin, ymax, invisible, tgrid, above = true); xaxis(BottomTop, xmin, xmax, invisible, tgrid, above = true); xaxis(xmin, xmax, Arrows(6), above = true); yaxis(ymin, ymax, Arrows(6), above = true);
draw((xmin,16/3)--(xmax,-2),red); limits((xmin,ymin),(xmax,ymax),Crop);
label("$x$", (xmax,0), E, fontsize(10)); label("$y$", (0,ymax), N, fontsize(10)); [/asy]

To find the linear inequality that describes the given solution set, we start by determining the equation of the line that forms the boundary of the shaded region. The line is depicted in red and crosses the y-axis and x-axis, giving us intercept points.

From the graph, we observe that:
- The line intersects the y-axis at $ \left(0, \frac{16}{3} \right) $.
- The line intersects the x-axis at $ (8, 0) $.

We can find the slope $ m $ of the line using these two points:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - \frac{16}{3}}{8 - 0} = \frac{-\frac{16}{3}}{8} = -\frac{16}{3 \times 8} = -\frac{2}{3}
\]

The slope-intercept form of the line equation is:
\[
y = mx + b
\]
Substituting the slope $ m $ and the y-intercept $ b = \frac{16}{3} $:
\[
y = -\frac{2}{3}x + \frac{16}{3}
\]

To convert this equation to standard form $ ax + by = c $, we first clear the fractions by multiplying through by 3:
\[
3y = -2x + 16
\]
Rearranging terms gives:
\[
2x + 3y = 16
\]

Next, we need to determine the correct inequality sign. Since the region shaded is above the line, we use the greater than sign. Therefore, the inequality is:
\[
2x + 3y \geq 16
\]

Thus, the linear inequality representing the solution set is:
\[
\boxed{2x + 3y \geq 16}
\]

Do you want details or have any questions?

1. What is the meaning of the slope in a linear equation?
2. How do you find the slope given two points?
3. What is the y-intercept and how is it determined from a graph?
4. How can you convert a line equation from slope-intercept form to standard form?
5. What does the shaded region in a graph of an inequality represent?
6. How do you decide whether to use a greater than or less than inequality?
7. What is the significance of multiplying through by a common factor to clear fractions?
8. How would you solve a system of inequalities graphically?

**Tip:** Always check whether the given points satisfy the final inequality to verify its correctness.

Learn how to find a linear inequality solution set using a graphical approach. Understand the concepts of slope, intercepts, and standard form of linear equations. Suitable for high school students studying algebra and geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation