Content

• Steps
• Method 1 of 3: Plotting Linear Inequality on the Number Line
• Method 2 of 3: Plotting Linear Inequality on a Coordinate Plane
• Method 3 of 3: Plotting Square Inequality on a Coordinate Plane
• Tips

The graph of a linear or square inequality is built in the same way as a graph of any function (equation) is built. The difference is that inequality implies multiple solutions, so an inequality graph is not just a point on a number line or a line on a coordinate plane. Using mathematical operations and the inequality sign, you can determine the set of solutions to the inequality.

Steps

Method 1 of 3: Plotting Linear Inequality on the Number Line

1. 1 Solve inequality. To do this, isolate the variable using the same algebraic techniques that you use to solve any equation. Remember that when multiplying or dividing an inequality by a negative number (or term), reverse the sign of the inequality.
   • For example, given the inequality ${ displaystyle 3y + 9> 12}$... To isolate the variable, subtract 9 from both sides of the inequality, and then divide both sides by 3:
     ${ displaystyle 3y + 9> 12}$
     ${ displaystyle 3y + 9-9> 12-9}$
     ${ displaystyle 3y> 3}$
     ${ displaystyle { frac {3y} {3}}> { frac {3} {3}}}$
     ${ displaystyle y> 1}$
   • Inequality must have only one variable. If the inequality has two variables, it is better to plot the graph on the coordinate plane.
2. 2 Draw a number line. On the number line, mark the value found (the variable can be less than, greater than or equal to this value). Draw a number line of the appropriate length (long or short).
   • For example, if you calculated that ${ displaystyle y> 1}$, on the number line mark the value 1.
3. 3 Draw a circle to represent the found value. If the variable is less (${ displaystyle}$) or more (${ displaystyle>}$) of this value, the circle is not filled, because many solutions do not include this value. If the variable is less than or equal to (${ displaystyle leq}$) or greater than or equal to (${ displaystyle geq}$) to this value, the circle is filled because many solutions include this value.
   • For example, given the inequality ${ displaystyle y> 1}$, on the number line, draw an open circle at point 1, because 1 is not included in the solution set.
4. 4 On the number line, shade the area that defines the set of solutions. If the variable is greater than the found value, shade the area to the right of it, because the solution set includes all values ​​that are greater than the found value. If the variable is less than the found value, shade the area to the left of it, because the solution set includes all values ​​that are less than the found value.
   • For example, given the inequality ${ displaystyle y> 1}$, on the number line, shade the area to the right of 1, because the set of solutions includes all values ​​greater than 1.

Method 2 of 3: Plotting Linear Inequality on a Coordinate Plane

1. 1 Solve inequality (find the value y{ displaystyle y}). To get a linear equation, isolate the variable on the left side using well-known algebraic methods. The variable should remain on the right side ${ displaystyle x}$ and possibly some constant.
   • For example, given the inequality ${ displaystyle 3y + 9> 9x}$... To isolate a variable ${ displaystyle y}$, subtract 9 from both sides of the inequality, and then divide both sides by 3:
     ${ displaystyle 3y + 9> 9x}$
     ${ displaystyle 3y + 9-9> 9x-9}$
     ${ displaystyle 3y> 9x-9}$
     ${ displaystyle { frac {3y} {3}}> { frac {9x-9} {3}}}$
     ${ displaystyle y> 3x-3}$
2. 2 Plot the linear equation on the coordinate plane. To do this, convert the inequality to an equation and plot the graph as you would any linear equation. Draw the y-intercept and then use the slope to add more points.
   • For example, in the case of inequality ${ displaystyle y> 3x-3}$ graph the equation ${ displaystyle y = 3x-3}$... The y-intercept has coordinates ${ displaystyle (0, -3)}$, and the slope is 3 (or ${ displaystyle { frac {3} {1}}}$). Thus, first draw a point with coordinates ${ displaystyle (0, -3)}$; the point above the y-intercept has coordinates ${ displaystyle (1,0)}$; the point below the y-intercept has coordinates ${ displaystyle (-1, -6)}$
3. 3 Draw a straight line. If the inequality is strict (includes the sign ${ displaystyle}$ or ${ displaystyle>}$), draw the dashed line, because the set of solutions does not include values ​​on the line. If the inequality is not strict (includes the sign ${ displaystyle leq}$ or ${ displaystyle geq}$), draw a solid line, because many solutions include values ​​that lie on a line.
   • For example, in the case of inequality ${ displaystyle y> 3x-3}$ draw a dashed line, because many solutions do not include values ​​on the line.
4. 4 Shade the appropriate area. If the inequality has the form ${ displaystyle y> mx + b}$, shade over the line. If the inequality has the form ${ displaystyle ymx + b}$, shade the area under the line.
   • For example, in the case of inequality ${ displaystyle y> 3x-3}$ shade over the line.

Method 3 of 3: Plotting Square Inequality on a Coordinate Plane

1. 1 Determine that the given inequality is square. The square inequality has the form ${ displaystyle ax ^ {2} + bx + c}$... Sometimes the inequality does not contain a first-order variable (${ displaystyle x}$) and / or a free term (constant), but necessarily includes a second-order variable (${ displaystyle x ^ {2}}$). Variables ${ displaystyle x}$ and ${ displaystyle y}$ must be isolated on different sides of inequality.
   • For example, you need to plot the inequality ${ displaystyle yx ^ {2} -10x + 16}$.
2. 2 Draw a graph on the coordinate plane. To do this, convert the inequality to an equation and plot the graph as you would any quadratic equation. Remember that the graph of a quadratic equation is a parabola.
   • For example, in the case of inequality ${ displaystyle yx ^ {2} -10x + 16}$ plot a quadratic equation ${ displaystyle y = x ^ {2} -10x + 16}$... The vertex of the parabola is at the point ${ displaystyle (5, -9)}$, and the parabola intersects the X-axis at points ${ displaystyle (2,0)}$ and ${ displaystyle (8.0)}$.
3. 3 Draw a parabola. If the inequality is strict (includes the sign ${ displaystyle}$ or ${ displaystyle>}$), draw a dashed parabola, because the solution set does not include the values ​​lying on the parabola. If the inequality is not strict (includes the sign ${ displaystyle leq}$ or ${ displaystyle geq}$), draw a solid parabola, because the set of solutions includes values ​​that lie on the parabola.
   • For example, in the case of inequality ${ displaystyle yx ^ {2} -10x + 16}$ draw a dotted parabola.
4. 4 Select some control points. To determine which area to shade, select the points inside and outside the parabola.
   • For example, on the graph of inequality ${ displaystyle yx ^ {2} -10x + 16}$ it can be seen that the point ${ displaystyle (0,0)}$ lies outside the parabola. This point can be used to define the area to be hatched.
5. 5 Shade the appropriate area. To determine which area to shade, substitute the values ${ displaystyle x}$ and ${ displaystyle y}$ control points. If, after substituting the coordinates of a point, the inequality is satisfied, shade the area in which this point lies.
   • For example, substitute the coordinate values ​​in the original inequality ${ displaystyle x}$ and ${ displaystyle y}$ points ${ displaystyle (0,0)}$:
     ${ displaystyle yx ^ {2} -10x + 16}$
     ${ displaystyle 00 ^ {2} -0x + 16}$
     ${ displaystyle 016}$
     Since the inequality is satisfied, shade the area in which the point lies ${ displaystyle (0,0)}$, that is, shade the area outside the parabola.

Tips

• Always simplify inequality before plotting it.
• If you cannot solve the problem, enter the inequality into a graphing calculator and try to solve the problem by working in the opposite direction.