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How To Solve Radical Inequalities Step by Step
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May 1, 2025
In this video, we build on our understanding of radical equations by learning how to solve radical inequalities — an essential concept inalgebra! 💡 I’ll walk you through the two-step process that makes radical inequalities different from standard equations, and explain why checking the domain is critical before solving. ✅ What You’ll Learn: What makes an inequality with a square root different from a standard inequality The two-step method for solving radical inequalities How to find the domain first (ensure the expression inside the radical is ≥ 0) How to solve the inequality by isolating the radical and squaring both sides How to combine intervals using a number line Why understanding inequality symbols matters (open vs closed circles)
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0:00
going to continue what we started
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yesterday but we're now going to talk
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about um solving inequalities that are
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radical right radical inequalities so
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the difference here is like two steps
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two steps when you have an inequality
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there's two things that you have to make
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sure actually the first thing is this
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you have to make sure the expression
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inside is
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positive right remember when we were
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graphing function we wanted you wanted
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to find the domain we want to make sure
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whatever is in here is greater than or
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equal to zero so before you solve this
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inequality we have to make sure that
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it's done so your very first step step
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one is to make sure that 5x in this case
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- 10 is what is greater than or equal to
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zero we have to make sure that is true
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first right which means you want to make
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sure that 5x is greater than or equal to
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10 and therefore x is greater than or
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equal to two that's step number one
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right before we even solve the
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inequality we have to make sure that we
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have this condition that is
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preset because if this is negative we
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can't solve the problem so we have to
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make sure that that the domain is
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properly defined right so we want to
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make sure that 5x - 10 is greater than
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or equal to 0 which means we want x to
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be greater than equal to 10 to two are
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you supposed to switch the side no
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because look at the question above
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doesn't matter i'm not done i'm not done
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oh you're not doing that one not yet the
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very first step right before you solve
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the problem you have to make sure when
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you have a radical that this radical
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here is greater than or equal to zero
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that's your first condition that has to
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be set okay now step two now we can
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solve the problem now we can actually
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solve the problem and once we finish the
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problem we're going to find the
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intersection of both intervals right so
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now we have here so I have
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three + 5x - 10 is less than or equal to
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8 right so what am I going to do now - 3
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- 3
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right so minus 3 so when we minus 3 we
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get
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5x - 10 is less than or equal to 5 and
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then what square square both sides right
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we going to square
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it and when you square it this takes
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care of that you left with 25 here and
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then we just solve it
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okay so we got 5x is less than or equal
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to
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35 and then x
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is less than or equal to 7 right so now
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we're going to do what we have the two
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conditions x has to be what x has to be
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greater than or equal to two and X also
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has to be what less than or equal to 7
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so if you're on the number line you're
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here right for X has to be first greater
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than or equal to two this is 2 and 7
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right x has to be greater than or equal
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to
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two and X has to be also what less than
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or equal to 7
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so where we made then
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between two and seven so your solution
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is what x has to be
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between
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2 and 7 that's your interval right so
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your solution will be 2 to 7 because
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these two conditions have to be met
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first X has to be greater than or equal
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to 2 and X also has to be less than or
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equal to 7 so if you do it Yes sir can
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you turn it up
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we can turn it off it's cold yeah just
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turn it off hit the up button and that's
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good all right so that's how we do it
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we'll do a couple more let's do a couple
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more problems here because this this can
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be a little bit confusing to some all
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right so now let me say I have this here
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right i have uh let me do another one
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2x + 2 + 1 is greater than or equal to 5
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right you see just make sure all right
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so what do we have to make sure first
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condition one what has to happen yeah
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wait sir is the the one is not under the
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clearly not no it's not yeah
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all right which means
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And then so x is one huh x is greater
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than or equal to one negative one you
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mean right that's condition one and now
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we have to solve this right so to solve
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it we have to go 2x +
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2 + 1 is more than equal to 5 i'm going
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to take out one on both side right so
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I'm going to get 2x +
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2 is greater than equal to 4 then what
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squaring squaring it right squaring
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is so we get 2x + 2 is greater than or
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equal to 16 / - 2 - 2 we get
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2x or equal to 14
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right and x is greater than equal to 7
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so I have two conditions running right
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now watch
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here so I want x to be greater than -1
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so I'm here right and I also want x to
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be what greater than equal to 7 i'm here
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so what do they mean
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uh maybe half right to seven so the
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solution be what seven to what infinity
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infinity wait when it's when it's
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greater than or equal to or less than
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equal to is it closed circle or open
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well closed circle is because if they're
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less than or equal to because of that
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right that that line under says that
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it's more than or equal or less than or
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equal so the solution will be from 7 to
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positive
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infinity all right let's do another one
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see I have um
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4 x - 4
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right - 2 that is less than strictly
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less than four so what has to happen
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first
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what's your first them you have to set
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it up to see so you have to do 4x - 4 is
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greater than or equal to zero all right
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which means x is greater than or equal
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to what four right
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and then x is greater than or equal to
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one right step and then next step you're
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going to do the the other one so step
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two we're going
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to add this right we're going to add two
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two so we gonna
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have is less than six then what
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square square you're going to square it
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all
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right squared so we got 4x - 4 is less
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than 36
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right and we add what
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four 4x is less than 40 and then x is
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less than 10 right so now watch we have
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two conditions here we have
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10 and one right x has to be what less
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than 10 no I'm sorry 10
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and one so X has to be more than one so
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more than one is right here going this
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direction and less than 10 is going this
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direction here so what's the solution
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one and 10 one and 10 so it's going to
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be one closed bracket and 10 what close
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bracket no is it open open open because
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it's less than yeah but it's going to be
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open or parenthesy all right so this is
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pretty much the last portion of this
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section it's a pretty short one so now
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we're just going to do some work in a
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book