Solving Radical Inequality Problems

Algebra: Inequalities Help


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I'll be honest. These are among the hardest algebra problems you'll ever have to solve in high school. There is a scene in the movie "Lawrence of Arabia" where they're crossing the desert. What they had to do is just slightly more difficult than solving these types of problems. The hardest part is keeping the logic straight. If you do that, you win the game. Rather than bore you with a bunch of theory, let's get right into the examples and I'll explain the logic as we go along.

By the way, you should know how to expand FOIL expressions, and use the quadratic formula to solve for the roots of the equation.

Example 1: Solve:

eq001

There are 2 logic hurdles we have to jump. The first is that you can't have the square root of a negative number. That means 3x - 2 > 0 or x > 2/3.
The second logic hurdle is this: since the left hand side is greater than or equal to zero and it is less than the right hand side, that means the right hand side must be greater than zero, which means 2x - 3 > 0 or x > 3/2. Combining these two conditions together, we must satisfy the condition of x > 3/2, which is the same as x > 1.5 converting the fraction to a decimal number. If we didn't take this into consideration before squaring both sides, we would end up with all sorts of problems.

The next step is to square both sides and bring everything over to one side.

3x - 2 < (2x - 3)2
3x - 2 < 4x2 - 12x + 9
4x2 - 15x + 11 > 0

And now for the next logic hurdle. Since this is a parabola, we need the parts of the parabola where y > 0. The first step is to solve for the roots of the equation using the quadratic formula. In this case, the roots are 1 and 2.75 This is made clear looking at the graph of the equation y = 4x2 - 15x + 11.

The parts of the parabola where y > 0 are x < 1 and x > 2.75.

But, if you recall from the beginning of the problem, we have the condition of x > 1.5. That eliminates the solution of x < 1. So the solution to our problem is x > 2.75.

The next example illustrates the point that keeping the logic straight not only helps you get the right answer but makes your life easier to boot.

Example 2: Solve:

eq002

Because the left hand side is greater than the right hand side, we have two cases to consider: 1) the right hand side is less than zero and 2) the right hand side is greater than zero.

As before, since we can't have the square root of a negative number, we have the condition 2x - 5 > 0 or x > 2.5 that must be satisfied.

Let's take the first case where the right hand side is less than zero. We do this one first because if the right hand side is less than zero, the left hand side is automatically greater since it is greater than or equal to zero. We have -4x + 3 < 0 or x > 0.75. Since we must satisfy the condition of x > 2.5 our solution is x > 2.5. Notice that we didn't have to square both sides to find the solution for this case. Isn't that nice?

For the second case where the right hand side is greater than zero, we have -4x + 3 > 0 or x < 0.75. But WE CAN'T HAVE x < 0.75 AND x > 2.5 AT THE SAME TIME. Ergo, there is no solution for the second case. The only solution for this problem is from the first case.

By the way, if we had gone ahead and blindly squared both sides, we would have ended up with 16x2 - 26x + 14 < 0. If we then tried to solve for the roots of this parabola, we would find that there aren't any and we would have reached the wrong conclusion that there is no solution to this problem. As I said before, using your head with these kinds of problems is a lot more important than just following an algorithm blindly.

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