Section2.6Linear Equations and Inequalities Chapter Review
Solving Multistep Linear Equations.
There is a regular process to use for solving a linear equation.
Simplify
Simplify the expressions on each side of the equation by distributing and combining like terms.
Separate
Use addition or subtraction to separate the terms so that the variable terms are on one side of the equation and the constant terms are on the other side of the equation.
Clear the Coefficient
Use multiplication or division to eliminate the variable termâs coefficient.
Check
Check the solution in the original equation. Substitute values into the original equation and use the order of operations to simplify both sides. Itâs important to use the order of operations alone rather than properties like the distributive law. Otherwise you might repeat the same arithmetic errors you (might have) made while solving, and fail to catch an incorrect solution.
Summarize
State the solution set. Or in the case of an application problem, summarize the result in a complete sentence using appropriate units.
Simplifying expressions, evaluating expressions, and solving equations are distinct algebra tasks.
An expression like \(10-3(x+2)\) can be simplified to \(-3x+4\) (as in Example 2.1.14). However we cannot âsolveâ an expression like this. It is incorrect to say you will âsolve \(10-3(x+2)\)â.
An expression like \(10-3(x+2)\) can be evaluated, but only once you have a number to use in place of the variable. This is what happens in Example 2.1.15, where \(x=2\text{,}\) and the expression evaluates to \(-2\text{.}\)
An equation connects two expressions with an equals sign. In Example 2.1.16, \(10-3(x+2)=x-16\) has one expression on either side of equals sign. You can solve this equation, because it is an equation. You can also solve inequalities. You just cannot solve an expression like \(10-3(x+2)\text{.}\)
When we solve the equation \(10-3(x+2)=x-16\text{,}\) we are looking for a number which makes those two expressions evaluate to the same value. In Example 2.1.16, we found the solution was \(5\text{.}\) That number \(5\) makes both \(10-3(x+2)\) and \(x-16\) evaluate to the same number (and that number is \(-11\) if you were curious.)
Checkpoint2.6.1.
Quotes are given from a student following a quiz. Fill in the blanks with the appropriate vocabulary terms.
(a)
The algebra quiz was fun because you had to
evaluate
simplify
solve
all these equations.
(b)
The last exercise seemed hard at first, but it became more clear once you
evaluated
simplified
solved
each side.
(c)
At first your answer to #4 might be pretty messy looking, but it looks more reasonable after you
evaluate
simplify
solve
it.
(d)
One of the word problems was asking you to take a given expression for the height of a ball and
evaluate
simplify
solve
it for \(t=7\text{.}\)
Checkpoint2.6.2.
\({-5j+8}={48}\)
Checkpoint2.6.3.
One of the tires on your car looks a little flat. You measure its air pressure and are alarmed to see it so low at \({24\ {\rm psi}}\text{.}\) You have a portable device that can pump air into the tire increasing the pressure at a rate of \({1.6\ {\textstyle\frac{\rm\mathstrut psi}{\rm\mathstrut min}}}\text{.}\) How long will it take to fill the tire to the manualâs recommended pressure of \({33\ {\rm psi}}\text{?}\)
Solving Multistep Linear Inequalities.
Solving a linear inequality is much like solving a linear equation. Two noteworthy differences are that multiplication/division by a negative number requires reversing the direction of the inequality symbol, and checking a solution takes more effort.
Simplify
Simplify the expressions on each side of the inequality by distributing and combining like terms.
Separate
Use addition or subtraction to separate the terms so that the variable terms are on one side of the inequality and the constant terms are on the other side of the inequality.
Clear the Coefficient
Use multiplication or division to eliminate the variable termâs coefficient. If you multiply or divide each side by a negative number, switch the direction of the inequality symbol.
Check
A solution to a linear inequality has a âboundary numberâ. Using the original inequality, check (1) a number less than the boundary number, (2) the boundary number itself, and (3) a number greater than the boundary number to confirm what should and shouldnât be solutions are all working as expected. (This can take time, so use your judgment about when you might get away with skipping this checking.)
Summarize
State the solution set. Or in the case of an application problem, summarize the result in a complete sentence using appropriate units.
Checkpoint2.6.4.
\({3v-2}\leq{-5}\)
Checkpoint2.6.5.
\({-8-5A}\leq{22}\)
Checkpoint2.6.6.
Leroy is driving on the highway, and presently has 18 gal of gasoline in his tank. His car, under ideal conditions, uses gas at a rate of 0.05 gal/mi. When the tank reaches only one gallon of gas, the low gas light will turn on and Leroy will start looking for a gas station. How far will he drive before this happens?
(a)
Write an inequality to represent this situation, using \(x\) to represent how many miles Leroy might drive before the low gas light turns on.
(b)
Solve this inequality. At most how far will Leroy drive before the low gas light turns on?
(c)
Use interval notation to express the number of miles Leroy might drive before the low gas light turns on.
Linear Equations and Inequalities with Fractions.
For both equations and inequalities, it is often helpful to âclear denominatorsâ if there are any fractions present. This is done by identifying the âleast common denominatorâ for the fractions that are present, and multiplying on each side by that number. This lets you avoid some fraction arithmetic that could lead to human error.
A proportional equation uses two ratios that should be equal to each other, for situations where two quantities change together. One example might use a ratio of how much of a substance is dissolved in how much of a certain liquid. Typically one ratio has two known numbers, and the other ration has one known number and one unknown variable.
An equation or inequality might reduce to an unambiguously true statement like \(2=2\) or \(4\lt9\text{.}\) If it does, then the soltuion set is all real numbers. This can be written as \((\infty,\infty)\) or \(\mathbb{R}\text{.}\)
Another special thing that can happen is that an equation or inequality can reduce to an unambiguously false statement like \(2=5\) or \(4\geq9\text{.}\) When this happens, then there is no solution at all. We can say that the solution set is âemptyâ. The solution set can be written as \(\{\}\) or \(\emptyset\text{.}\)
Isolating a Linear Variable.
Equations might relate two or more variables to eachother. When you have an equation like that (for example \(2x+3y=4z\)) you can solve for any one of the variables, isolating it on one side of an equals sign. The example could be solved for \(y\text{,}\) which gives \(y=\frac{4z-2x}{3}\text{.}\)
The process for doing this is not different than solving a one-variable equation. You should just keep clear which variable you are solving for.
ExercisesReview Exercises for Chapter 2
Section 1: Solving Multistep Linear Equations
1.
This exercise walks you through an economics problem. Fill in the blanks with the appropriate vocabulary terms.
(a)
If the revenue from selling \(x\) items is \(2x\) dollars but the cost of production is \(0.5x\text{,}\) then the profit is \(2x-0.5x\text{.}\) But that can be
evaluated
simplified
solved
.
(b)
We need to know how much profit there was after \(1000\) items, so we can
evaluate
simplify
solve
the profit expression.
(c)
If we want to work out how many sales would give us a profit of \(\$3000\text{,}\) we can
evaluate
simplify
solve
the equation \(1.5x=3000\text{.}\)
(d)
At first we find that it would take \(\frac{3000}{1.5}\) sales, but we can
You planted a young tree in front of your house, and it was \(5\) feet tall. Ever since, it has been growing by \({{\frac{11}{12}}\ {\rm ft}}\) each year. How many years will it take for the tree to grow to be 12 feet tall?
10.
Amiyah puts a pot of water from the tap onto the stove and turns the burner all the way up. The water temperature starts at \(56\,â\) and climbs steadily up to the boiling point of \(212\,â\text{,}\) raising at a rate of \(24\,\frac{â}{\text{min}}\text{.}\) How long will it take for the pot to boil?
11.
On a cold snowy day, the temperature in your home is a cozy \(70\,â\text{,}\) but then you lose power and heating. Your home temperature begins to drop at a rate of \(11\,\frac{â}{\text{hour}}\text{.}\) How long will it take before your home is \(44\,â\text{?}\)
12.
For Eliasâs 8th birthday party, his parents rented a venue that charges a flat fee of \({\$100}\) plus \({\$13}\) per guest. Utimately it cost Eliasâs parents \({\$347}\text{.}\) How many guests were there?
13.
A dishwasher is for sale in a state where sales tax applies. The sales tax rate is \({5.2\%}\) and the total was \({\$430}\text{.}\) What was the price before sales tax?
14.
One year, the median rent for a one-bedroom apartment in a city was reported to be \({\$1{,}580}\text{.}\) This was reported to be an increase of \({2\%}\) over the previous year. Based on this reporting, what was the median rent for of a one-bedroom apartment the previous year?
