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2.09 Inequalities with variables on both sides

Lesson

Solving more complicated inequalities

We have looked at how to solve one-step, two-step and multi-step inequalities.  In this lesson, we will look at solving more complicated inequalities - like those that have variables on both sides of the inequality symbol.  

Worked example

question 1

Solve the inequality for $x$x:   $4(2x+6)\le4x-8$4(2x+6)4x8

Think:  In order to isolate the variable in this inequality, I will need to first remove the parenthesis and move one of the x terms - so that the variables are all on the same side of the inequality symbol.

Do:  

$4(2x+6)$4(2x+6) $\le$ $4x-8$4x8

 

$8x+24$8x+24 $\le$ $4x-8$4x8

Distribute the $4$4 to remove the parenthesis

$8x-4x+24$8x4x+24 $\le$ $4x-4x-8$4x4x8

Subtract $4x$4x from both sides of the inequality 

$4x+24$4x+24 $\le$ $-8$8

Now, we will use the reverse order of operations to isolate $x$x

$4x+24-24$4x+2424 $\le$ $-8-24$824

Subtract $24$24 from both sides of the inequality

$4x$4x $\le$ $-32$32

Then, divide both sides by $4$4

$x$x $\le$ $-8$8  

 

Reflect:  Solving inequalities with variables on both sides of the inequality symbol is very much like solving equations with variables on both sides.  The thing we want to keep in mind while working is: If while we are solving we multiply or divide by a negative number, we must reverse the direction of the inequality symbol.  

Here are some more complicated examples for you to practice solving more challenging inequalities.  

Practice questions

QUESTION 2

Solve the inequality $\frac{2x+5}{9}-4\ge5$2x+5945.

QUESTION 3

Solve the inequality $\frac{x+7}{5}+3x<9$x+75+3x<9.

QUESTION 4

Consider the inequality $8x-11>5x+4$8x11>5x+4.

  1. Solve the inequality.

  2. Hence, plot the inequality $8x-11>5x+4$8x11>5x+4 on the number line below.

    -10-50510

Outcomes

8.18

Solve multistep linear inequalities in one variable with the variable on one or both sides of the inequality symbol, including practical problems, and graph the solution on a number line

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