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.
Check the solution in the original equation. Substitute values into the original equation and use the order of operations to simplify both sides. It’s 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.
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 “solve” an expression like this. It is incorrect to say you will “solve \(10-3(x+2)\)”.
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 equal sign. In Example 2.1.16, \(10-3(x+2)=x-16\) has one expression on either side of equal 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.)
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 \({20\ {\rm psi}}\text{.}\) You have a portable device that can pump air into the tire increasing the pressure at a rate of \({1.8\ {\textstyle\frac{\rm\mathstrut psi}{\rm\mathstrut min}}}\text{.}\) How long will it take to fill the tire to the manual’s recommended pressure of \({33\ {\rm psi}}\text{?}\)
This is a rate scenario. The air pressure starts out at \(20\text{,}\) and increases with a rate of \(1.8\text{.}\) Eventually the pressure reaches \(33\text{.}\) So the equation we set up is:
\begin{equation*}
20+1.8t=33
\end{equation*}
where \(t\) is the number of minutes the pump runs. Now we follow the standard steps to solve this equation.
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.
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.
Use multiplication or division to eliminate the variable term’s coefficient. If you multiply or divide each side by a negative number, switch the direction of the inequality symbol.
A solution to a linear inequality has a “boundary number”. 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 shouldn’t 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.)
We can graph this by shading all numbers greater than \(4\text{.}\) Since the sign in the inequality does not allow for \(z\) to equal \(4\text{,}\) we use a parenthesis at \(4\text{.}\)
We can graph this by shading all numbers less than \(-1\text{.}\) Since the sign in the inequality does not allow for \(G\) to equal \(-1\text{,}\) we use a parenthesis at \(-1\text{.}\)
Miguel is driving on the highway, and presently has 12 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 Miguel will start looking for a gas station. How far will he drive before this happens?
The question mark in the above is not an equal sign, because we are trying to set up an inequality. The initial value for the gas in his tank is \(12\text{,}\) Miguel is losing gas (so we will subtract, not add), the rate is \(0.05\text{,}\) and the gas amount we are interested in reaching is \(1\text{.}\) We do not know far Miguel can drive until the gas gets that low, and we decide to use \(x\) to represent that distance. The rate of gas consumption we are using is only valid under ideal conditions. So the quantity \(12 - 0.05 x\) will be greater than or equal to the actual amount of gas left in the tank after driving \(x\) miles. And we want this to reach \(1\text{.}\) So the inequality we set up is:
Miguel can drive at most \({220}\) miles, so the right end of the interval notation for this interval will be \({220}\text{,}\) with a bracket. We know Miguel has more than 1 gallon of gas when this all started, so he can drive at least some positive number of miles. So the interval notation, accounting for the context of this exercise, is \({\left(0,220\right]}\text{.}\)
For both equations and inequalities, it is often helpful to “clear denominators” if there are any fractions present. This is done by identifying the “least common denominator” 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 ratio has one known number and one unknown variable.
This equation has terms with denominators \(4\) and \(11\text{.}\) The least common multiple of these is \({44}\text{.}\) So the first step in solving this equation can be to multiply on each side by \({44}\text{.}\)
This inequality has terms with denominators \({30}\text{,}\)\({42}\text{,}\) and \({35}\text{.}\) The least common multiple of these is \({210}\text{.}\) So the first step in solving this inequality can be to multiply on each side by \({210}\text{.}\)
The solution set is \(\{x\mid x \leq {{\frac{73}{7}}}\}\) using set-builder notation. In interval notation, we have \({\left(-\infty ,{\frac{73}{7}}\right]}\text{.}\)
An old cookbook has a recipe for soup that uses \(10\) g of flour for thickener. The recipe makes \(1\) gal of soup, but you are adjusting it to make only \(1.25\) gal. How much flour should you use?
Let \(x\) be how many grams of flour we should use. The amount of flour is proportional to the amount of soup, so we can compare the ratios of flour to soup for our batch with the original recipe:
These denominators are not whole numbers, and we decide not to try using the least common multiple. Instead, we just multiply on each side by both denominators:
\begin{equation*}
\begin{aligned}
1\cdot 1.25\cdot \frac{x}{1.25} \amp= 1\cdot 1.25\cdot \frac{10}{1} \\
1 x \amp= 12.5\\
x \amp= \frac{12.5}{1 }\\
x \amp= 12.5
\end{aligned}
\end{equation*}
An equation or inequality might reduce to an unambiguously true statement like \(2=2\) or \(4\lt9\text{.}\) If it does, then the solution 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 “empty”. The solution set can be written as \(\{\}\) or \(\emptyset\text{.}\)
This is an outright true equation, so all real numbers are solutions. The solution set is \((-\infty,\infty)\text{,}\) which we can write as \(\mathbb{R}\text{.}\)
Equations might relate two or more variables to each other. 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 equal sign. The example could be solved for \(y\text{,}\) which gives \(y=\frac{4z-2x}{3}\text{.}\)
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
You planted a young tree in front of your house, and it was \(6\) feet tall. Ever since, it has been growing by \({{\frac{3}{4}}\ {\rm ft}}\) each year. How many years will it take for the tree to grow to be 11 feet tall?
Ricky puts a pot of water from the tap onto the stove and turns the burner all the way up. The water temperature starts at \(58\,^{\circ}F\) and climbs steadily up to the boiling point of \(212\,^{\circ}F\text{,}\) raising at a rate of \(29\,\frac{^{\circ}F}{\text{min}}\text{.}\) How long will it take for the pot to boil?
On a cold snowy day, the temperature in your home is a cozy \(70\,^{\circ}F\text{,}\) but then you lose power and heating. Your home temperature begins to drop at a rate of \(15\,\frac{^{\circ}F}{\text{hour}}\text{.}\) How long will it take before your home is \(36\,^{\circ}F\text{?}\)
For Alyssa’s 8th birthday party, her parents rented a venue that charges a flat fee of \({\$115}\) plus \({\$16}\) per guest. Ultimately it cost Alyssa’s parents \({\$435}\text{.}\) How many guests were there?
A car stereo is for sale in a state where sales tax applies. The sales tax rate is \({4.5\%}\) and the total was \({\$314}\text{.}\) What was the price before sales tax?
One year, the median rent for a one-bedroom apartment in a city was reported to be \({\$1{,}160}\text{.}\) This was reported to be an increase of \({1.5\%}\) over the previous year. Based on this reporting, what was the median rent for of a one-bedroom apartment the previous year?
You hail a taxi and can only pay with cash. This cab service charges a flat fee of \({\$15.00}\) and then charges \({\$3.50}\) per mile. No tip is expected. You are carrying a total of \({\$106.00}\) in cash with you. You want to know how many miles you can afford.
The population of a certain country grew by \(6\%\) over the course of the past decade. One city in this country grew in population too, but at an even faster rate than the country grew overall. The city’s current population is \(86344\text{.}\) What might its population have been ten years ago? (Note: the population ten years ago is known to have been at least \(10{,}000\text{.}\))
You planted a young tree in front of your house, and it was \(4\) feet tall. Ever since, it has been growing by \({{\frac{1}{6}}\ {\rm ft}}\) each year. How many years will it take for the tree to grow to be 11 feet tall?
In one college math course, the final grade (out of \(100\) points) is calculated by adding one-half of the homework average, one-sixth of the midterm score, and one-third of the final exam score.
Edgar has 97 for his homework average and 90 for his midterm score. His goal is to finish the course with a final grade of 70. What must he score on the final exam?
Garrison and Jayleen went to the weekend market to buy apples in bulk for a school event, canning lots of applesauce. Garrison bought 58 lb of apples and paid $67.28. Jayleen has bagged her apples and has 90 lb of apples ready. How much will it Jayleen’s bag of apples cost?
To estimate the health of the black-tailed deer population in the Jewell Meadow Wildlife Area, the Oregon Department of Fish and Wildlife caught, tagged, and released \(90\) black-tailed deer. A month later, they returned and observed \(79\) black-tailed deer, \(11\) of which had tags. Approximately how many black-tailed deer are in the Jewell Meadow Wildlife Area?
