Name: graphing inequality solution ranges
Uploaded: 2025-11-26T05:44:15.000Z
Duration: 1 min 24 s
Channel: JoVE

通过图像法求解不等式需要选择 x 值，利用函数计算相应的 y 值，并绘制图像。该图显示了突出显示满足不等式的 x 值范围的区域。在同一坐标系中，绘制另一个不等式函数，并阴影标记不等式成立的边界线一侧。重叠区域中的 x 值表示该不等式的解。对于二次函数，当不等式为大于或等于时，图像的阴影部分表示其解；类似地，当不等式为小于或等于时，解对应于图像的非阴影区域。当二次不等式与线性函数进行比较时，两个图像绘制在同一坐标平面上。图像在若干点处相交，这些 x 值标出了解集的边界，表明抛物线位于直线的上方或与直线重合的区域。用图形法求解不等式在金融等领域具有实用价值，例如可用于比较每月支出与预算。阴影区域表明支出始终低于预算线。

用图解法求解不等式

Solving Inequalities Graphically

&#29992;&#22270;&#35299;&#27861;&#27714;&#35299;&#19981;&#31561;&#24335;&#26159;&#19968;&#31181;&#36890;&#36807;&#22270;&#24418;&#21270;&#26041;&#24335;&#30830;&#23450;&#25968;&#23398;&#34920;&#36798;&#24335;&#28385;&#36275;&#29305;&#23450;&#26465;&#20214;&#65288;&#22914;&#22823;&#20110;&#25110;&#23567;&#20110;&#26576;&#19968;&#20540;&#65289;&#30340;&#26041;&#27861;&#12290;&#36890;&#36807;&#35266;&#23519;&#22270;&#20687;&#30456;&#23545;&#20110; x &#36724;&#25110;&#21478;&#19968;&#20989;&#25968;&#30340;&#22270;&#20687;&#30340;&#20301;&#32622;&#65292;&#21487;&#20197;&#30830;&#23450;&#28385;&#36275;&#19981;&#31561;&#24335;&#30340; x &#20540;&#33539;&#22260;&#12290;&#36825;&#31181;&#26041;&#27861;&#36890;&#36807;&#30452;&#35266;&#23637;&#31034;&#19981;&#31561;&#24335;&#25104;&#31435;&#30340;&#21306;&#22495;&#65292;&#20174;&#32780;&#24110;&#21161;&#29702;&#35299;&#35299;&#38598;&#30340;&#21306;&#38388;&#33539;&#22260;&#12290;
&#22270;&#35299;&#27861;&#27714;&#35299;&#19981;&#31561;&#24335;&#36890;&#24120;&#38656;&#35201;&#32472;&#21046;&#30456;&#20851;&#20989;&#25968;&#65292;&#24182;&#30830;&#23450;&#20986;&#19981;&#31561;&#24335;&#26465;&#20214;&#25104;&#31435;&#30340;&#21306;&#38388;&#12290;&#20363;&#22914;&#65292;&#23545;&#20110;&#20108;&#27425;&#19981;&#31561;&#24335; x^2 - 4x + 2 &lt; 0&#65292;&#32472;&#21046;&#20989;&#25968; y = x^2 - 4x + 2&#65292;&#20854;&#35299;&#20026;&#26354;&#32447;&#20301;&#20110; x &#36724;&#19979;&#26041;&#30340;&#25152;&#26377; x &#20540;&#65292;&#21363;&#19981;&#31561;&#24335;&#25104;&#31435;&#30340;&#21306;&#38388;&#12290;
&#33509;&#28041;&#21450;&#20004;&#20010;&#20989;&#25968;&#30340;&#19981;&#31561;&#24335;&#65292;&#22914; x^3 - 4x^2 + 8 &#8805; -8&#65292;&#21017;&#38656;&#20998;&#21035;&#32472;&#21046;&#20989;&#25968; y_1 = x^3 - 4x^2 + 8 &#21644; y_2 = -8&#12290;&#35299;&#38598;&#30001;&#25152;&#26377;&#28385;&#36275; y_1 &#8805; y_2 &#30340; x &#20540;&#32452;&#25104;&#65292;&#21363; y_1 &#30340;&#22270;&#20687;&#20301;&#20110; y_2 &#30340;&#22270;&#20687;&#20043;&#19978;&#25110;&#19982;&#20043;&#37325;&#21512;&#30340;&#21306;&#22495;&#12290;
&#22270;&#35299;&#27861;&#33021;&#22815;&#30452;&#35266;&#22320;&#26174;&#31034;&#35299;&#21306;&#38388;&#65292;&#19981;&#20165;&#20415;&#20110;&#24555;&#36895;&#21028;&#26029;&#19981;&#31561;&#24335;&#30340;&#35299;&#65292;&#36824;&#33021;&#24110;&#21161;&#29702;&#35299;&#20989;&#25968;&#30340;&#24615;&#36136;&#19982;&#30456;&#20114;&#20851;&#31995;&#12290;

Education

教育

JoVE Core

Math Fundamentals

数学基础

Unit

单位

Coordinates and Graphs

KEY TERMS AND CONCEPTS

关键术语与概念

graphing inequality solution ranges

Graphing inequality solution ranges gives a visual way to find the x-values that make an inequality true. Students look at where a graph sits compared with the x-axis or with another graph. This shows the interval where the condition is satisfied.

For a quadratic inequality, the first step is to plot the related function. For example, with x² − 4x + 2 &lt; 0, graph y = x² − 4x + 2. The solution is the set of x-values where the curve lies below the x-axis.

Inequalities can also compare two functions. For x³ − 4x² + 8 ≥ −8, graph y₁ = x³ − 4x² + 8 and y₂ = −8. The answer includes all x-values where y₁ is on or above y₂. Graphical methods make these solution intervals easy to spot and help show the shape of the function.

Graphing Inequality Solution Ranges

coordinate plane

Finding Points on the Coordinate Plane

The Cartesian coordinate plane is used to find and plot points in two dimensions. It is made of two number lines that cross at right angles. The horizontal line is the x-axis, and the vertical line is the y-axis. Their meeting point is the origin, where both values are zero. The axes divide the plane into four quadrants. These quadrants are labeled in a counterclockwise direction, starting from the upper right. Every point on the plane is written as an ordered pair. The first number is the...

the distance formula

Finding Distance on a Coordinate Plane

Finding distance on a coordinate plane uses the distance formula and the Pythagorean Theorem. In geometry, this method helps measure the direct distance between two points on a flat plane. It is useful in many practical and theoretical settings. The formula comes from a right triangle. On a coordinate plane, the horizontal and vertical differences between two points become the legs of that triangle. The line connecting the points is the hypotenuse, and that is the shortest path between them. <br...

the midpoint formula

Finding the Center of a Line Segment

Finding the center of a line segment is a key idea in coordinate geometry. The midpoint is the point that lies exactly halfway between two points on a coordinate plane. It is used in mathematics, physics, engineering, and planning work where a central location matters. To work with a midpoint, start with two points, A(x1, y1) and B(x2, y2). A straight line segment can be drawn between them on the coordinate plane. The midpoint, labeled M, divides that segment into two equal parts. The...

graphs of equations in two variables

Plotting Two-Variable Equations with Tables

Plotting two-variable equations with tables helps show how x and y are related. An equation written as y = f(x) or Ax + By = C gives ordered pairs that satisfy the rule. Each pair can be checked by substituting the x and y values into the equation. A value table is a common way to build the graph. Start by choosing several values for the independent variable x. Then substitute each x value into the equation to find the matching y value. The results become ordered pairs on the Cartesian...

circles

Circle Equations in Standard Form

Circle equations in the coordinate plane use a center and a radius to describe every point on the curve. A circle is the set of all points that stay the same distance from a fixed point called the center. That constant distance is the radius. The distance formula connects a point x, y on the circle to the center h, k. The distance between those two points equals the radius r. When both sides of the distance formula are squared, the equation of the circle can be written in standard form. <br...

symmetry

Ellipse Symmetry at the Origin

An ellipse centered at the origin shows clear symmetry in its graph. Its equation describes a smooth, closed curve whose distance from the center keeps a constant ratio between the horizontal and vertical axes. Because the ellipse stretches farther along the x-axis than the y-axis, it has a horizontal shape. This ellipse has symmetry across the x-axis. That means the top half and bottom half are mirror images across the horizontal axis. Algebraically, the equation stays the same when y is...

solving equations graphically

Graphing Equations to Find Intercepts

Graphing equations helps students find intercepts and solutions on the coordinate plane. It gives a visual way to solve equations and understand how a function behaves. To use this method, write the equation in the form y = f(x). The solution to the original equation is the x-value where the graph crosses the x-axis. At that point, f(x) = 0. For example, the linear equation 2x - 4 = 0 can be written as y = 2x - 4. When this function is graphed, it has one x-intercept at x = 2, and that is the...

solving inequalities graphically

Graphing inequality solution ranges gives a visual way to find the x-values that make an inequality true. Students look at where a graph sits compared with the x-axis or with another graph. This shows the interval where the condition is satisfied. For a quadratic inequality, the first step is to plot the related function. For example, with x² − 4x + 2 &lt; 0, graph y = x² − 4x + 2. The solution is the set of x-values where the curve lies below the x-axis. Inequalities can also compare two...

key terms and concepts

Use graphs to find the x-values that satisfy inequalities. Learn how to solve quadratic and two-function inequalities by checking where curves lie on the graph.

Graphing Inequality Solution Ranges | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Graphing Inequality Solution Ranges

Description

Transcript

Tags