In this lesson you will yet again be building off of previous ones, this time with doing a different form of the one and two-step algebraic equations you have delt with in 1.3 and 1.5. However, rather than there being an equal sign, there will be in an inequality sign, such as the greater than symbol (>), less than symbol (<), greater than or equal to symbol (≥), or less than or equal to symbol (≤). Additionally, the meaning of x will be different from a normal equation. For example, if you get the answer x>5, then that means any number above 5 will satisfy the inequality, whether it be 5.001, 100, or even 12,000,000. One final note: if you have to divide or multiply by a negative number, the symbol will have to be flipped, such as > flipping to be < or ≤ flipping to ≥. This occurs to keep the inequality true, due to multiplying/dividing by a negative basically reversing the order of numbers that would be seen on a number line. While there is not much new in this lesson, solving inequalities will be important later on, and it is still good to refine one and two-step equations skills.Now, let us get into some examples. As usual, examples (and the problem set) will get more difficult as it goes on, so make sure to do every one to slowly shift into more difficult questions.
1: x+15>19
Starting off easy, this is very similar to a one-step equation where all that has to be done to solve the inequality is simply apply the inverse operation of addition (which is subtraction), and subtract 15 from both sides to get the answer of x>4.
Answer: x>4
2: x·12≤72
In this one, the inverse operation of multiplication (division) must be used. Diving by 12 on both sides yields the answer by cancelling the 12 out on the left side and dividing 72 on the right to get the answer of x≤6.
Answer: x≤6
3: ˣ⁄₋₇<21
In this inequality, x is being divided, so there is one step on top of simply multiplying. by doing the inverse operation of division and multiplying both sides by -7, we get the answer x<-147.
However, since the inequality has a negative number multiplying each side, we have to flip the inequality sign, making the answer x>-147.
Also, if you want to see why the symbol has to be flipped, try making the inequality true by plugging a number into x that is great than -147 (aka something like -146).
Answer: x>-147
4: -10x≥300
Firstly, the inverse operation of multiplication (division) must be done, dividing both sides by -10 to get x≥-30.
Yet again, since the inequality was divided by a negative number, the sign must be flipped, giving the final answer of x≤-30.
Answer: x≤-30
5: 10x+3<19.5
Now we are moving into more of what two-step equations would be if they were imagined as inequalities. First, the inverse operation of addition should be done, subtracting 3 from both sides and making the equality 10x<16.5
Secondly, the inverse operation of multiplication will be done, dividing 16.5 by 10 to get the final of x<1.65.
Answer: x<1.65
6: ˣ⁄₅+5≥-9.5
First, the inverse operation of addition (subtraction) must be done, subtracting 5 from both sides giving the inequality ˣ⁄₅≥-14.5.
Secondly, the inverse operation of division should be done, multiplying both sides of the inequality by 5 to get the final answer of x≥-72.5.
Answer: x≥-72.5
7: -4.5-6x≤33
In this inequality, -4.5-6x can also be written as -6x-4.5, but either way the inverse operation of subtraction should be applied both sides, adding 4.5 to get -6x≤37.5.
Secondly, the inverse operation of multiplication should be done, dividing 37.5 by -6 to get the final answer of -6.25, but due to dividing by a negative the sign has to be flipped, making it x≥-6.25.
Answer: x≥-6.25
8: 8+√x<17
First of all, the inverse operation of addition should be done, subtracting 8 from 17 to get the difference of 9.
Secondly, the inverse of square rooting something (squaring) should be done, squaring 9 to get the final answer of x<9.
Answer: x<9
9: ∛x-17>-12
First and foremost, the inverse operation of subtraction must be done, then adding 17 both sides to get ∛x>5.
Secondly, the cube root must be cancelled by cubing both sides of the equation, giving the final answer of x>125.
Answer: x>125
10: ˣ⁄₋₃-9≥0
As usual, the inverse operation of subtraction must be applied first and foremost, making the inequality ˣ⁄₋₃≥9.
Secondly, the inverse operation of division must be utlizied, multiplying both sides by -3, flipping the inequality sign and giving the final answer of x≤-27.
Answer: x≤-27
11: -4+x³≤23
First, the inverse operation of subtraction must be applied, adding 4 to both sides and making the equality x³≤27.
Secondly, the inverse operation of cubing something (cube root) must be applied, giving the final answer of x≤3.
Answer: x≤3
12: -33x-33≥-333
Even when there are larger numbers, still follow the same protocol/steps. First, the inverse operation of subtraction should be applied by adding 33 to each side and making the inequality -33x≥-300.
Secondly, the inverse operation of multiplication must be applied, flipping the inequality symbol and giving us the final answer of x≤9.1.
Answer: x≤9.1
1: 2x+15<-30
2: ˣ⁄₋₃-4≤-8
3: -33+2x≥99
4: ˣ⁄₂+13≥49
5: x³-24<3
6: ˣ⁄₋₁₁-8>22
7: x³-22≥64
8: -8+√x>4.5
9: 0+10x<-10.5
10: -4.5+x²≤15
11: 12-3x≥-19
12: -8x+73≤-9
13: -4x+91>22
14: 4+x²≤23
15: -5x+6<43
16: -4-4.9x≤8.45
17: ∛x+2>-6
18: -13+∛x≥-20
1: x<-22.5
2: x≥12
3: x≥66
4: x≥72
5: x<3
6: x<-330
7: x≥4.41
8: x>156.25
9: x<-2.05
10: x≤4.42
11: x≤10.33
12: x≥10.25
13: x<17.25
14: x≤4.36
15: x>-7.4
16: x≥-2.54
17: x>-512
18: x≥ -343