UK Secondary (7-11)
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$3x−5 $<$< $-5x+11$−5x+11 $8x-5$8x−5 $<$< $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.

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

##### Question 2

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

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

$y=4x+1$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