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Section2.2Equations and Inequalities as True/False Statements

This section introduces the concepts of algebraic equations and inequalities, and what it means for a number to be a solution to an equation or inequality.

Figure2.2.1Alternative Video Lesson

Subsection2.2.1Equations, Inequalities, and Solutions

An equation is two mathematical expressions with an equals sign between them. The two expressions can be relatively simple or more complicated:

A relatively simple equation:

\begin{equation*} x+1=2 \end{equation*}

A more complicated equation:

\begin{equation*} \left(x^2+y^2-1\right)^3=x^2y^3 \end{equation*}

An inequality is quite similar, but the sign between the expressions is one of these: \(\lt\text{,}\) \(\leq\text{,}\) \(\gt\text{,}\) \(\geq\text{,}\) or \(\neq\text{.}\)

A relatively simple inequality:

\begin{equation*} x\geq15 \end{equation*}

A more complicated inequality:

\begin{equation*} x^2+y^2\lt1 \end{equation*}

A linear equation in one variable can be written in the form \(ax+b=0\text{,}\) where \(a, b\) are real numbers, and \(a\ne0\text{.}\) The variable could be any letter other than \(x\text{.}\) The variable cannot have other exponent than \(1\) (\(x=x^1\)), and the variable cannot be inside a root symbol (square root, cube root, etc.) or in a denominator.

The following are some linear equations in one variable:

\begin{align*} 4-y\amp=5 \amp 4-z\amp=5z \amp 0\amp=\frac{1}{2}p \end{align*} \begin{align*} 3-2(q+2)\amp=10 \amp \sqrt{2}\cdot r+3\amp=10 \amp \frac{s}{2}+3\amp=5 \end{align*}

(Note that \(r\) is outside the square root symbol.) We will see in later sections that all equations above can be converted into the form \(ax+b=0\text{.}\)

The following are some non-linear equations:

\begin{align*} 1+2=3 \amp\amp\amp \text{(There is no variable.)}\\ 4-2y^2=5 \amp\amp\amp \text{(The exponent of $y$ is not $1$.)}\\ \sqrt{2r}+3=10 \amp\amp\amp \text{($r$ is inside the square root.)}\\ \frac{2}{s}+3=5 \amp\amp\amp \text{($s$ is in a denominator.)} \end{align*}

This chapter focuses on linear equations in one variable. We will study other types of equations in later chapters.

The simplest equations and inequalities have numbers and no variables. When this happens, the equation is either true or false.The following equations and inequalities are true statements:

\begin{align*} 2\amp=2\amp-4\amp=-4\amp2\amp\gt1\amp-2\amp\lt-1\amp3\amp\ge3 \end{align*}

The following equations and inequalities are false statements:

\begin{align*} 2\amp=1\amp-4\amp=4\amp2\amp\lt1\amp-2\amp\ge-1\amp0\neq0 \end{align*}

When equations and inequalities have variables, we can consider substituting values in for the variables. If replacing a variable with a number makes an equation or inequality true, then that number is called a solution to the equation.

Example2.2.2A Solution

Consider the equation \(y+2=3\text{,}\) which has only one variable, \(y\text{.}\) If we substitute in \(1\) for \(y\) and then simplify:

\begin{align*} y+2\amp=3\\ \substitute{1}+2\amp\stackrel{?}{=}3\\ 3\amp\stackrel{\checkmark}{=}3 \end{align*}

we get a true equation. So we say that \(1\) is a solution to \(y+2=3\text{.}\) Notice that we used a question mark at first because we are unsure if the equation is true or false until the end.

If replacing a variable with a number makes an equation or inequality false, then that number is not a solution.

Example2.2.3Not a Solution

Consider the inequality \(x+4\gt 5\text{,}\) which has only one variable, \(x\text{.}\) If we substitute in \(0\) for \(x\) and then simplify:

\begin{align*} x+4\amp\gt 5\\ \substitute{0}+4\amp\stackrel{?}{\gt}5\\ 4\amp\stackrel{\text{no}}{\gt}5 \end{align*}

we get a false equation. So we say that \(0\) is not a solution to \(x+4\gt 5\text{.}\)

Subsection2.2.2Checking Possible Solutions

Given an equation or an inequality (with one variable), checking if some particular number is a solution is just a matter of replacing the value of the variable with the specified number and determining if the resulting equation/inequality is true or false. This may involve some amount of arithmetic simplification.

Example2.2.4

Is \(8\) a solution to \(x^2-5x=\sqrt{2x}+20\text{?}\)

To find out, substitute in \(8\) for \(x\) and see what happens.

\begin{align*} x^2-5x\amp=\sqrt{2x}+20\\ \substitute{8}^2-5(\substitute{8})\amp\stackrel{?}{=}\sqrt{2(\substitute{8})}+20\\ \highlight{64}-5(8)\amp\stackrel{?}{=}\sqrt{\highlight{16}}+20\\ 64-\highlight{40}\amp\stackrel{?}{=}\highlight{4}+20\\ \highlight{24}\amp\stackrel{\checkmark}{=}\highlight{24} \end{align*}

So yes, \(8\) is a solution to \(x^2-5x=\sqrt{2x}+20\text{.}\)

Example2.2.5

Is \(-5\) a solution to \(\sqrt{169-y^2}=y^2-2y\text{?}\)

To find out, substitute in \(-5\) for \(y\) and see what happens.

\begin{align*} \sqrt{169-y^2}\amp=y^2-2y\\ \sqrt{169-\substitute{(-5)}^2}\amp\stackrel{?}{=}\substitute{(-5)}^2-2(\substitute{-5})\\ \sqrt{169-\highlight{25}}\amp\stackrel{?}{=}\highlight{25}-2(-5)\\ \sqrt{\highlight{144}}\amp\stackrel{?}{=}25-(\highlight{-10})\\ \highlight{12}\amp\stackrel{\text{no}}{=}\highlight{35} \end{align*}

So no, \(-5\) is not a solution to \(\sqrt{169-y^2}=y^2-2y\text{.}\)

But is \(-5\) a solution to the inequality \(\sqrt{169-y^2}\leq y^2-2y\text{?}\) Yes, because substituting \(-5\) in for \(y\) would give you

\begin{equation*} 12\leq35\text{,} \end{equation*}

which is true.

Exercise2.2.6
Exercise2.2.7
Exercise2.2.8
Exercise2.2.9
Exercise2.2.10
Example2.2.11Cylinder Volume

A cylinder's volume is related to its radius and its height by the equation

\begin{equation*} V=\pi r^2h\text{,} \end{equation*}

where \(V\) is the volume, \(r\) is the base's radius, and \(h\) is the height. If we know the volume is 96\(\pi\)‚ÄĮcm3 and the radius is 4‚ÄĮcm, then this equation simplifies to

\begin{equation*} 96\pi=16\pi h \end{equation*}
a right cylinder with radius labeled 4 cm; its volume is labeled as 96 pi cm^3
Figure2.2.12A cylinder

Is 4‚ÄĮcm the height of the cylinder? In other words, is \(4\) a solution to \(96\pi=16\pi h\text{?}\) We will substitute \(h\) in the equation with \(4\) to check:

\begin{align*} 96\pi\amp=16\pi h\\ 96\pi\amp\stackrel{?}{=}16\pi \cdot\substitute{4}\\ 96\pi\amp\stackrel{\text{no}}{=}64\pi \end{align*}

Since \(96\pi=64\pi\) is false, \(h=4\) does not satisfy the equation \(96\pi=16\pi h\text{.}\)

Next, we will try \(h=6\text{:}\)

\begin{align*} 96\pi\amp=16\pi h\\ 96\pi\amp\stackrel{?}{=}16\pi \cdot\substitute{6}\\ 96\pi\amp\stackrel{\checkmark}{=}96\pi \end{align*}

When \(h=6\text{,}\) the equation \(96\pi=16\pi h\) is true. This tells us that \(6\) is a solution for \(h\) in the equation \(96\pi=16\pi h\text{.}\)

Remark2.2.13

Note that we did not approximate \(\pi\) with \(3.14\) or any other approximation. We often leave \(\pi\) as \(\pi\) throughout our calculations. If we need to round, we do so as a final step.

Example2.2.14

Ann has budgeted a maximum of \(\$300\) for an appliance repair. The total cost of the repair can be modeled by \(89+110(h-0.25)\text{,}\) where \(\$89\) is the initial cost and \(\$110\) is the hourly labor charge after the first quarter hour. Is \(2\) a solution for \(h\) in the inequality \(89+110(h-0.25)\le 300\text{?}\)

To determine if \(h=2\) satisfies the inequality \(89+110(h-0.25)\le 300\text{,}\) we will replace \(h\) with \(2\) and check if the statement is true or false:

\begin{align*} 89+110(h-0.25)\amp\le 300\\ 89+110(\substitute{2}-0.25)\amp\stackrel{?}{\le} 300\\ 89+110(1.75)\amp\stackrel{?}{\le} 300\\ 89+192.5\amp\stackrel{?}{\le} 300\\ 281.5\amp\stackrel{\checkmark}{\le} 300 \end{align*}

Thus \(2\) is a solution for \(h\) in the inequality \(89+110(h-0.25)\le 300\text{.}\) In context, this means that Ann would stay within her \(\$300\) budget if \(2\) hours of labor were performed.

Subsection2.2.3Exercises

Checking Solution for Equations

1

Is \(7\) a solution for \(x\) in the equation \({x-6} = {-13}\text{?}\)

?

Yes

No

2

Is \(-5\) a solution for \(x\) in the equation \({x+8} = {3}\text{?}\)

?

Yes

No

3

Is \(2\) a solution for \(y\) in the equation \({4-y} = {3}\text{?}\)

?

Yes

No

4

Is \(-6\) a solution for \(y\) in the equation \({-8-y} = {-2}\text{?}\)

?

Yes

No

5

Is \(8\) a solution for \(r\) in the equation \({r-9} = {-2}\text{?}\)

?

Yes

No

6

Is \(10\) a solution for \(r\) in the equation \({6r+2} = {62}\text{?}\)

?

Yes

No

7

Is \(7\) a solution for \(t\) in the equation \({-3t+1} = {2t+36}\text{?}\)

?

Yes

No

8

Is \(6\) a solution for \(t\) in the equation \({6t-6} = {t+24}\text{?}\)

?

Yes

No

9

Is \(-2\) a solution for \(x\) in the equation \({3\!\left(x+7\right)} = {-4x}\text{?}\)

?

Yes

No

10

Is \(-7\) a solution for \(x\) in the equation \({7\!\left(x-5\right)} = {12x}\text{?}\)

?

Yes

No

11

Is \(5\) a solution for \(y\) in the equation \({5\!\left(y+1\right)} = {-6\!\left(y-10\right)}\text{?}\)

?

Yes

No

12

Is \(2\) a solution for \(y\) in the equation \({8\!\left(y-9\right)} = {11\!\left(y-6\right)}\text{?}\)

?

Yes

No

13

Is \({{\frac{1}{9}}}\) a solution for \(x\) in the equation \({-9x+8} = 7\text{?}\)

?

Yes

No

14

Is \({{\frac{1}{2}}}\) a solution for \(x\) in the equation \({-6x+5} = 1\text{?}\)

?

Yes

No

15

Is \({-{\frac{4}{7}}}\) a solution for \(r\) in the equation \({-{\frac{3}{4}}r+4} = {{\frac{25}{7}}}\text{?}\)

?

Yes

No

16

Is \({-{\frac{1}{7}}}\) a solution for \(t\) in the equation \({{\frac{5}{4}}t+10} = {{\frac{275}{28}}}\text{?}\)

?

Yes

No

17

Is \({-{\frac{9}{5}}}\) a solution for \(t\) in the equation \({{\frac{4}{3}}t - {\frac{1}{5}}} = {-{\frac{1}{4}}t - {\frac{61}{20}}}\text{?}\)

?

Yes

No

18

Is \({-{\frac{3}{5}}}\) a solution for \(x\) in the equation \({-{\frac{4}{5}}x+1} = {-{\frac{2}{3}}x+{\frac{27}{25}}}\text{?}\)

?

Yes

No

Checking Solution for Inequalities

19

Tell whether the following values are solutions to the given inequality.

\(2 x - 11 > -7\)

  1. \(x=7\)

    ?

    is

    is not

    a solution.

  2. \(x=2\)

    ?

    is

    is not

    a solution.

  3. \(x=0\)

    ?

    is

    is not

    a solution.

  4. \(x=-4\)

    ?

    is

    is not

    a solution.

20

Tell whether the following values are solutions to the given inequality.

\(3 x - 3 > 9\)

  1. \(x=0\)

    ?

    is

    is not

    a solution.

  2. \(x=14\)

    ?

    is

    is not

    a solution.

  3. \(x=4\)

    ?

    is

    is not

    a solution.

  4. \(x=-3\)

    ?

    is

    is not

    a solution.

21

Tell whether the following values are solutions to the given inequality.

\(-3 x +9 \ge 3\)

  1. \(x=7\)

    ?

    is

    is not

    a solution.

  2. \(x=2\)

    ?

    is

    is not

    a solution.

  3. \(x=0\)

    ?

    is

    is not

    a solution.

  4. \(x=-5\)

    ?

    is

    is not

    a solution.

22

Tell whether the following values are solutions to the given inequality.

\(3 x - 15 \ge -3\)

  1. \(x=-3\)

    ?

    is

    is not

    a solution.

  2. \(x=14\)

    ?

    is

    is not

    a solution.

  3. \(x=0\)

    ?

    is

    is not

    a solution.

  4. \(x=4\)

    ?

    is

    is not

    a solution.

23

Tell whether the following values are solutions to the given inequality.

\(4 x - 13 \le -9\)

  1. \(x=6\)

    ?

    is

    is not

    a solution.

  2. \(x=0\)

    ?

    is

    is not

    a solution.

  3. \(x=-6\)

    ?

    is

    is not

    a solution.

  4. \(x=1\)

    ?

    is

    is not

    a solution.

24

Tell whether the following values are solutions to the given inequality.

\(-4 x +18 \le 6\)

  1. \(x=0\)

    ?

    is

    is not

    a solution.

  2. \(x=3\)

    ?

    is

    is not

    a solution.

  3. \(x=-4\)

    ?

    is

    is not

    a solution.

  4. \(x=13\)

    ?

    is

    is not

    a solution.

Checking Solutions for Application Problems

25

A triangle’s area is \(171\) square meters. Its height is \(19\) meters. Suppose we wanted to find how long is the triangle’s base. A triangle’s area formula is

\(\displaystyle{A=\frac{1}{2}bh}\text{,}\)

where \(A\) stands for area, \(b\) for base and \(h\) for height. If we let \(b\) be the triangle’s base, in meters, we can solve this problem using the equation:

\(\displaystyle{{171}=\frac{1}{2}(b)(19)}\text{.}\)

Check whether \(36\) is a solution for \(b\) of this equation.

?

Yes

No

26

A triangle’s area is \(150\) square meters. Its height is \(15\) meters. Suppose we wanted to find how long is the triangle’s base. A triangle’s area formula is

\(\displaystyle{A=\frac{1}{2}bh}\text{,}\)

where \(A\) stands for area, \(b\) for base and \(h\) for height. If we let \(b\) be the triangle’s base, in meters, we can solve this problem using the equation:

\(\displaystyle{{150}=\frac{1}{2}(b)(15)}\text{.}\)

Check whether \(20\) is a solution for \(b\) of this equation.

?

Yes

No

27

A cylinder’s volume is \(160\pi\) cubic centimeters. Its height is \(10\) centimeters. Suppose we wanted to find how long is the cylinder’s radius. A cylinder’s volume formula is

\(\displaystyle{V=\pi r^2h}\text{,}\)

where \(V\) stands for volume, \(r\) for radius and \(h\) for height. Let \(r\) represent the cylinder’s radius, in centimeters. We can solve this problem using the equation:

\(\displaystyle{160 \pi=\pi r^2(10)}\text{.}\)

Check whether \(16\) is a solution for \(r\) of this equation.

?

Yes

No

28

A cylinder’s volume is \(704\pi\) cubic centimeters. Its height is \(11\) centimeters. Suppose we wanted to find how long is the cylinder’s radius. A cylinder’s volume formula is

\(\displaystyle{V=\pi r^2h}\text{,}\)

where \(V\) stands for volume, \(r\) for radius and \(h\) for height. Let \(r\) represent the cylinder’s radius, in centimeters. We can solve this problem using the equation:

\(\displaystyle{704 \pi=\pi r^2(11)}\text{.}\)

Check whether \(8\) is a solution for \(r\) of this equation.

?

Yes

No

29

A rectangular frame’s perimeter is \(7\) feet. If its length is \(2.2\) feet, suppose we want to find how long is its width. A rectangle’s perimeter formula is

\(\displaystyle{P=2(l+w)}\text{,}\)

where \(P\) stands for perimeter, \(l\) for length and \(w\) for width. We can solve this problem using the equation:

\(\displaystyle{7=2(2.2+w)}\text{.}\)

Check whether \(4.8\) is a solution for \(w\) of this equation.

?

Yes

No

30

A rectangular frame’s perimeter is \(6.8\) feet. If its length is \(2.3\) feet, suppose we want to find how long is its width. A rectangle’s perimeter formula is

\(\displaystyle{P=2(l+w)}\text{,}\)

where \(P\) stands for perimeter, \(l\) for length and \(w\) for width. We can solve this problem using the equation:

\(\displaystyle{6.8=2(2.3+w)}\text{.}\)

Check whether \(4.5\) is a solution for \(w\) of this equation.

?

Yes

No

31

When a plant was purchased, it was \(2.6\) inches tall. It grows \(0.6\) inches per day. How many days later will the plant be \(11\) inches tall?

Assume the plant will be \(11\) inches tall \(d\) days later. We can solve this problem using the equation:

\(\displaystyle{0.6 d+2.6=11}\text{.}\)

Check whether \(14\) is a solution for \(d\) of this equation.

?

Yes

No

32

When a plant was purchased, it was \(1.9\) inches tall. It grows \(0.7\) inches per day. How many days later will the plant be \(14.5\) inches tall?

Assume the plant will be \(14.5\) inches tall \(d\) days later. We can solve this problem using the equation:

\(\displaystyle{0.7 d+1.9=14.5}\text{.}\)

Check whether \(18\) is a solution for \(d\) of this equation.

?

Yes

No

33

A water tank has \(210\) gallons of water in it, and it is being drained at the rate of \(16\) gallons per minute. After how many minutes will there be \(34\) gallons of water left?

Assume the tank will have \(34\) gallons of water after \(m\) minutes. We can solve this problem using the equation:

\(\displaystyle{210-16 m=34}\text{.}\)

Check whether \(11\) is a solution for \(m\) of this equation.

?

Yes

No

34

A water tank has \(349\) gallons of water in it, and it is being drained at the rate of \(17\) gallons per minute. After how many minutes will there be \(43\) gallons of water left?

Assume the tank will have \(43\) gallons of water after \(m\) minutes. We can solve this problem using the equation:

\(\displaystyle{349-17 m=43}\text{.}\)

Check whether \(18\) is a solution for \(m\) of this equation.

?

Yes

No

35

A country’s national debt was \(140\) million dollars in 2010. The debt increased at \(70\) million dollars per year. If this trend continues, when will the country’s national debt increase to \(910\) million dollars?

Assume the country’s national debt will become \(910\) million dollars \(y\) years after 2010. We can solve this problem using the equation:

\(\displaystyle{70 y+140=910}\text{.}\)

Check whether \(11\) is a solution for \(y\) of this equation. (This solution implies the country’s national debt will become \(910\) million dollars in the year 2021.)

?

Yes

No

36

A country’s national debt was \(100\) million dollars in 2010. The debt increased at \(20\) million dollars per year. If this trend continues, when will the country’s national debt increase to \(480\) million dollars?

Assume the country’s national debt will become \(480\) million dollars \(y\) years after 2010. We can solve this problem using the equation:

\(\displaystyle{20 y+100=480}\text{.}\)

Check whether \(17\) is a solution for \(y\) of this equation. (This solution implies the country’s national debt will become \(480\) million dollars in the year 2027.)

?

Yes

No

37

A school district has a reserve fund worth \(33.9\) million dollars. It plans to spend \(2.1\) million dollars per year. After how many years, will there be \(15\) million dollars left?

Assume there will be \(15\) million dollars left after \(y\) years. We can solve this problem using the equation:

\(\displaystyle{33.9-2.1 y=15}\text{.}\)

Check whether \(9\) is a solution for \(y\) of this equation.

?

Yes

No

38

A school district has a reserve fund worth \(26.4\) million dollars. It plans to spend \(2.2\) million dollars per year. After how many years, will there be \(11\) million dollars left?

Assume there will be \(11\) million dollars left after \(y\) years. We can solve this problem using the equation:

\(\displaystyle{26.4-2.2 y=11}\text{.}\)

Check whether \(7\) is a solution for \(y\) of this equation.

?

Yes

No