topic badge
India
Class XI

Solve Linear Inequalities from a Graph

Lesson

Previously, we learnt how to solve linear inequalities. Remember that a linear inequality in $x$x will not involve $x^2$x2, $x^3$x3, $\frac{1}{x}$1x, or anything else. It will only ever involve $x$x being multiplied or divided by constants, or having constants added or subtracted.

In other words, the left hand side and right hand side of a linear equality will both be lines!

Think about it. If we have the linear inequality $3x-5<-5x+11$3x5<5x+11, the left hand side is the line $y_1=3x-5$y1=3x5 and the right hand side is the line $y_2=-5x+11$y2=5x+11.

We could solve the inequality algebraically like this.

$3x-5$3x5 $<$< $-5x+11$5x+11  
$8x-5$8x5 $<$< $11$11 Add $5x$5x to both sides
$8x$8x $<$< $16$16 Add $5$5 to both sides
$x$x $<$< $2$2 Divide both sides by $8$8.

Or, we could realise that to solve $3x-5<-5x+11$3x5<5x+11 we just need to find for what values of $x$x the line $y_1=3x-5$y1=3x5 is less than $y_2=-5x+11$y2=5x+11.

We can graph both lines like this.

For what values of $x$x is the line $y_1=3x-5$y1=3x5 less than $y_2=-5x+11$y2=5x+11? In other words, when is the line $y_1=3x-5$y1=3x5 below the line $y_2=-5x+11$y2=5x+11?

Just as we found by solving the inequality algebraically, $y_1$y1 is below $y_2$y2 for $x<2$x<2.

Worked Examples

Question 1

Using the above graph, state the solution to the linear inequality $3x+9\ge0$3x+90 and plot the solution on a number line.

The solution to the inequality $3x+9\ge0$3x+90 will be whenever the line $y=3x+9$y=3x+9 is above the line $y=0$y=0 (the $x$x-axis), in other words, when the line is positive.

We can see that this happens when $x\ge-3$x3, which we can plot on the number line like this.

Question 2

Using the above graph, state the solution to the linear inequality $3x-21<9$3x21<9 and plot the solution on a number line.

 

The solution to the inequality $3x-21<9$3x21<9 will be whenever the line $y=3x-21$y=3x21 is below the line $y=9$y=9.

We can see that this happens when $x<10$x<10, which we can plot on the number line like this.

 

Worked Examples

Question 1

Consider the graph of $y=x+6$y=x+6.

Loading Graph...

  1. Using the graph, state the solution of the inequality $x+6<0$x+6<0.

    Give your answer in interval notation.

Question 2

Consider the graphs of $y=-23$y=23 and $y=4x-3$y=4x3.

 

Loading Graph...
Two lines $StraightLine(CartesianCoordinate(-6,-27,'',True),CartesianCoordinate(-5,-23,'',False))$StraightLine(CartesianCoordinate(6,27,,True),CartesianCoordinate(5,23,,False)) and $StraightLine(CartesianCoordinate(0,-23,'',True),CartesianCoordinate(1,-23,'',True))$StraightLine(CartesianCoordinate(0,23,,True),CartesianCoordinate(1,23,,True)) graphed on a coordinate plane.
  1. Using the graphs, state the solution of the inequality $4x-3<-23$4x3<23. Give your answer in interval notation.

Question 3

To solve the inequality $x\le\frac{x-3}{4}-1$xx341, Tracy graphed $y=x-3$y=x3. What other line would she need to graph to be able to solve the inequality graphically?

  1. $y=4x+1$y=4x+1

    A

    $y=\frac{x-3}{4}-1$y=x341

    B

    $y=x$y=x

    C

    $y=4x+4$y=4x+4

    D

Outcomes

11.A.LE.1

Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical solution of linear inequalities in two variables. Solution of system of linear inequalities in two variables – graphically.

What is Mathspace

About Mathspace