in this lesson we're going to focus on solving inequalities and at the same time graphing it on a number line so let's begin with our first example 2x plus 3 is greater than 7. so to solve an inequality is the same as solving an equation you can treat the greatest sign as if it's an equal sign the only difference is when you have the solution you should plot it on a number line so let's begin by subtracting both sides by 3. these will cancel and so 2x is greater than 7 minus 3 which is 4. next let's divide both sides by 2. and so we can see that x is greater than 4 divided by two which is two and now let's plot it on the number line so here's two because it's greater than but not equal to we're gonna have an open circle and since it's greater than we're gonna shade towards the right that is towards positive infinity so to represent the answer using interval notation it's from 2 to infinity now let's try another example try this one 1 3 x plus 4 is less than or equal to 6. feel free to pause the video and work on this example now let's begin by subtracting 4 from both sides so 1 3x is less than or equal to six minus four which is two next let's get rid of the fraction so let's multiply both sides by three three times one third is one so we're going to be left with one x or just x two times three is six so x is less than or equal to six so this time we're going to have a closed circle and because it's less than we're going to shade towards the left that is towards negative infinity so to represent the answer using interval notation it's from negative infinity to 6 but because we have a closed circle at six we're going to use brackets but keep in mind you should always have parentheses anytime you're dealing with infinity let's try another example go ahead and find the solution to this equation and then plot it on a number line so what do you think the first thing we should do in this example should we divide by negative 3 or subtract by five what you want to do is subtract by five first these will cancel negative three x will be equal to a greater than negative four minus five which is negative nine next we need to divide both sides by negative three there's a question for you when you divide or multiply by a negative number does anything special happens and the answer is yes when you multiply or divide by a negative number you need to change the direction of the inequality sign negative nine divided by negative three is positive three and now we can plot the solution on a number line so x is less than or equal to three so we're going to have a closed circle shaded towards the left because it's less than so the solution using interval notation is negative infinity to three using brackets here's the next one let's say if we have two x minus one is greater than seven or negative three x plus two is equal to or greater than negative one solve the equations so let's start with the first one let's add one to both sides so two x is greater than eight next let's divide both sides by two eight divided by two is four so x is greater than four now before we plot our number line let's solve the other one let's subtract both sides by two negative one minus two is negative three next let's divide by negative three don't forget to change the direction of the inequality sign so this is what we have so there's a one actually let's put zero first here's zero one two three and then four so x is greater than four but not equal to four so we're gonna have an open circle at four shaded towards the right now x is less than or equal to one so we're gonna have a closed circle shaded towards the left since it's less than so using interval notation the answer is negative infinity to 1 using brackets union 4 to infinity and that's it let's try one more example let's say negative 12 is less than 7x minus 5 which is less than or equal to 9. find a solution so here we have a compound inequality with three sides to an equation so let's add five to all three sides negative 12 plus five is negative seven negative five plus five is zero nine plus five is fourteen next let's divide each side by seven negative seven divided by seven is negative one seven x divided by seven is just x 14 divided by 7 is 2. so x is between negative 1 and 2. x is greater than 1 but not equal to it so we're going to have an open circle shaded towards the right x is less than or equal to two so we're going to have a closed circle shaded towards the left so it's between these two values so an interval notation is from negative infinity to two so you have a bracket which is associated with a closed circle and a parenthesis that is associated with the open circle