Examples of Solving Harder Linear Inequalities
Purplemath
Once you'd learned how to solve one-variable linear equations, you were then given word problems. To solve these problems, you'd have to figure out a linear equation that modelled the situation, and then you'd have to solve that equation to find the answer to the word problem.
So, now that you know how to solve linear inequalities — you guessed it! — they give you word problems.
- The velocity of an object fired directly upward is given by V = 80 − 32t, where the time t is measured in seconds. When will the velocity be between 32 and 64 feet per second (inclusive)?
This question is asking when the velocity, V, will be between two given values. So I'll take the expression for the velocity,, and put it between the two values they've given me. They've specified that the interval of velocities is inclusive, which means that the interval endpoints are included. Mathematically, this means that the inequality for this model will be an "or equal to" inequality. Because the solution is a bracket (that is, the solution is within an interval), I'll need to set up a three-part (that is, a compound) inequality.
I will set up the compound inequality, and then solve for the range of times t:
Note that, since I had to divide through by a negative, I had to flip the inequality signs.
Note also that you might (as I do) find the above answer to be more easily understood if it's written the other way around, with "less than" inequalities.
And, because this is a (sort of) real world problem, my working should show the fractions, but my answer should probably be converted to decimal form, because it's more natural to say "one and a half seconds" than it is to say "three-halves seconds". So I convert the last line above to the following:
0.5 ≤ t ≤ 1.5
Looking back at the original question, it did not ask for the value of the variable "t", but asked for the times when the velocity was between certain values. So the actual answer is:
The velocity will be between 32 and 64 feet per second between 0.5 seconds after launch and 1.5 seconds after launch.
Okay; my answer above was *extremely* verbose and "complete"; you don't likely need to be so extreme. You can probably safely get away with saying something simpler like, "between 0.5 seconds and 1.5 seconds". Just make sure that you do indeed include the approprioate units (in this case, "seconds").
Always remember when doing word problems, that, once you've found the value for the variable, you need to go back and re-read the problem to make sure that you're answering the actual question. The inequality 0.5 ≤ t ≤ 1.5 did not answer the actual question regarding time. I had to interpret the inequality and express the values in terms of the original question.
- Solve 5x + 7 < 3(x + 1).
First I'll multiply through on the right-hand side, and then solve as usual:
5x + 7 < 3(x + 1)
5x + 7 < 3x + 3
2x + 7 < 3
2x < −4
x < −2
In solving this inequality, I divided through by a positive 2 to get the final answer; as a result (that is, because I did *not* divide through by a minus), I didn't have to flip the inequality sign.
- You want to invest $30,000. Part of this will be invested in a stable 5%-simple-interest account. The remainder will be "invested" in your father's business, and he says that he'll pay you back with 7% interest. Your father knows that you're making these investments in order to pay your child's college tuition with the interest income. What is the least you can "invest" with your father, and still (assuming he really pays you back) get at least $1900 in interest?
First, I have to set up equations for this. The interest formula for simple interest is I = Prt, where I is the interest, P is the beginning principal, r is the interest rate expressed as a decimal, and t is the time in years.
Since no time-frame is specified for this problem, I'll assume that t = 1; that is, I'll assume (hope) that he's promising to pay me at the end of one year. I'll let x be the amount that I'm going to "invest" with my father. Then the rest of my money, being however much is left after whatever I give to him, will be represented by "the total, less what I've already given him", so 30000 − x will be left to invest in the safe account.
Then the interest on the business investment, assuming that I get paid back, will be:
I = (x)(0.07)(1) = 0.07x
The interest on the safe investment will be:
(30 000 − x)(0.05)(1) = 1500 − 0.05x
The total interest is the sum of what is earned on each of the two separate investments, so my expression for the total interest is:
0.07x + (1500 − 0.05x) = 0.02x + 1500
I need to get at least $1900; that is, the sum of the two investments' interest must be greater than, or at least equal to, $1,900. This allows me to create my inequality:
0.02x + 1500 ≥ 1900
0.02x ≥ 400
x ≥ 20 000
That is, I will need to "invest" at least $20,000 with my father in order to get $1,900 in interest income. Since I want to give him as little money as possible, I will give him the minimum amount:
I will invest $20,000 at 7%.
- An alloy needs to contain between 46% copper and 50% copper. Find the least and greatest amounts of a 60% copper alloy that should be mixed with a 40% copper alloy in order to end up with thirty pounds of an alloy containing an allowable percentage of copper.
This is similar to a mixture word problem, except that this will involve inequality symbols rather than "equals" signs. I'll set it up the same way, though, starting with picking a variable for the unknown that I'm seeking. I will use x to stand for the pounds of 60% copper alloy that I need to use. Then 30 − x will be the number of pounds, out of total of thirty pounds needed, that will come from the 40% alloy.
Of course, I'll remember to convert the percentages to decimal form for doing the algebra.
| pounds | % copper | pounds copper | |
|---|---|---|---|
| 60% alloy |
x | 0.6 | 0.6x |
| 40% alloy |
30 − x | 0.4 | 0.4(30 − x) = 12 − 0.4x |
| mixture | 30 | between 0.46 and 0.5 |
between 13.8 and 15 |
How did I get those values in the bottom right-hand box? I multiplied the total number of pounds in the mixture (30) by the minimum and maximum percentages (46% and 50%, respectively). That is, I multiplied across the bottom row, just as I did in the "60% alloy" row and the "40% alloy" row, to get the right-hand column's value.
The total amount of copper in the mixture will be the sum of the copper contributed by each of the two alloys that are being put into the mixture. So I'll add the expressions for the amount of copper from each of the alloys, and place the expression for the total amount of copper in the mixture as being between the minimum and the maximum allowable amounts of copper:
13.8 ≤ 0.6x + (12 − 0.4x) ≤ 15
13.8 ≤ 0.2x + 12 ≤ 15
1.8 ≤ 0.2x ≤ 3
9 ≤ x ≤ 15
Checking back to my set-up, I see that I chose my variable to stand for the number of pounds that I need to use of the 60% copper alloy. And they'd only asked me for this amount, so I can ignore the other alloy in my answer.
I will need to use between 9 and 15 pounds of the 60% alloy.
Per yoozh, I'm verbose in my answer. You can answer simply as "between 9 and 15 pounds".
- Solve 3(x − 2) + 4 ≥ 2(2x − 3)
First I'll multiply through and simplify; then I'll solve:
3(x − 2) + 4 ≥ 2(2x − 3)
3x − 6 + 4 ≥ 4x − 6
3x − 2 ≥ 4x − 6
−2 ≥ x − 6 (*)
4 ≥ x
x ≤ 4
Why did I move the 3x over to the right-hand side (to get to the line marked with a star), instead of moving the 4x to the left-hand side? Because by moving the smaller term, I was able to avoid having a negative coefficient on the variable, and therefore I was able to avoid having to remember to flip the inequality when I divided through by that negative coefficient. I find it simpler to work this way; I make fewer errors. But it's just a matter of taste; you do what works for you.
Why did I switch the inequality in the last line and put the variable on the left? Because I'm more comfortable with inequalities when the answers are formatted this way. Again, it's only a matter of taste. The form of the answer in the previous line, 4 ≥ x, is perfectly acceptable.
As long as you remember to flip the inequality sign when you multiply or divide through by a negative, you shouldn't have any trouble with solving linear inequalities.
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