ORCCA Equations and Inequalities as True/False Statements

Section 1.4 Equations and Inequalities as True/False Statements

Objectives: PCC Course Content and Outcome Guide

MTH 60 CCOG 2.1
MTH 60 CCOG 2.2

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.

Figure 1.4.1. Alternative Video Lesson

Subsection 1.4.1 Equations, Inequalities, and Solutions

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

A 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 similar to an equation, but the sign between the expressions is $\lt\text{,}$ $\leq\text{,}$ $\gt\text{,}$ $\geq\text{,}$ or $\neq\text{.}$

A simple inequality:

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

A more complicated inequality:

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

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*}

There will be times when doing algebra will lead us to an equation like $2=1\text{,}$ which of course we know to be a false equation. To recognize that this is false, we will write $1\stackrel{\text{no}}{=}2\text{.}$ This is different from writing $1\neq2\text{,}$ because that is a true inequality. And when we want to explicitly recognize that an equation or inequality is true, we will use a checkmark, like with $2\stackrel{\checkmark}{=}2\text{.}$

A linear expression in one variable is an expression in the form $ax+b\text{,}$ where $a$ and $b$ are numbers, $a\neq0\text{,}$ and $x$ is a variable. For example, $2x+1$ and $3y+\frac{1}{2}$ are linear expressions.

The following examples are a little harder to identify as linear expressions in one variable, but they are.

$2x$ is linear, with $b=0\text{.}$
$y+3$ is linear, with $a=1\text{.}$
$17-q$ is linear, with $a=-1\text{,}$ $b=17$ and the two terms are written in reverse order.
$2.1t+3+8t-1.4$ is linear (because it simplifies to $10.1t+1.6$).

Definition 1.4.2. Linear Equation and Linear Inequality.

A linear equation in one variable is any equation where one side is a linear expression in that variable, and the other side is either a constant number, or is another linear expression in that variable. A linear inequality in one variable is defined similarly, just with an inequality symbol instead of an equals sign.

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} r+3\amp=10 \amp \frac{s}{2}+3\amp=5 \end{align*}

(Note that $r$ is outside the square root symbol.)

In a linear equation in one variable, the variable cannot appear with an exponent (other than $1$ or $0$), and the variable cannot be inside a root symbol (square root, cube root, etc.), absolute value bars, or in a denominator.

The following are not linear equations in one variable:

\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 $2$.)}\\ \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*}

Equations arise from real-world math problems, sometimes from simple problems, and sometimes from hard ones.

Example 1.4.3.

A parking meter requires you pay $2.50 for one hour. You have been inserting quarters, dimes, and nickels into the meter, and it says that you have inserted $1.85. How much more do you need to pay?

You might have a simple way to answer that question, using subtraction. But there is an equation hidden in this story. Since we are asked “How much more do you need to pay?”, let's use a variable to represent that: $x\text{.}$ We've already paid $1.85, and in total we need to pay $2.50. So we need

\begin{equation*} 1.85+x=2.50 \end{equation*}

This is an equation arising from this scenario.

With the equation in Example 1.4.3, if we substitute in $0.65$ for $x\text{,}$ the resulting equation is true.

\begin{equation*} 1.85+0.65\stackrel{\checkmark}{=}2.50 \end{equation*}

If we substitute in any other number for $x\text{,}$ the resulting equation is false. This motivates what it means to be a solution to an equation.

Definition 1.4.4.

When an equation (or inequality) has one variable, a solution is any number that you could substitute in for the variable that would result in a true equation (or inequality).

Example 1.4.5. A 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 value makes a false equation or inequality, that number is not a solution.

Example 1.4.6. Not 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 inequality. So we say that $0$ is not a solution to $x+4\gt 5\text{.}$

Example 1.4.7. Allowing Variables to Vary.

With the help of technology, it is possible to quickly evaluate expressions as variables vary. In the GeoGebra applet in Figure 1.4.8, you may slide the value of $q$ and see how a computer can quickly calculate each side of the equation to determine if that value of $q$ is a solution.

Figure 1.4.8. Allowing Variables to Vary

Subsection 1.4.2 Checking 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 arithmetic and simplification.

Example 1.4.9.

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 $x^2-5x=\sqrt{2x}+20\text{.}$

Example 1.4.10.

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.

Checkpoint 1.4.11.

Checkpoint 1.4.12.

Checkpoint 1.4.13.

Checkpoint 1.4.14.

Checkpoint 1.4.15.

Example 1.4.16. Cylinder Volume.

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

\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$\pi$ cm³ and the radius is 4 cm, then we have:

\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

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Is 4 cm 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 to $96\pi=16\pi h\text{.}$

Remark 1.4.17.

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.

Example 1.4.18.

Jaylen 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 $2$ hours a solution for $h$ in the inequality $89+110(h-0.25)\le 300\text{?}$

To determine if $h=2$ satisfies the inequality, we will replace $h$ with $2$ and check if the statement is true:

\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*}

So we find that $2$ is a solution for $h$ in the inequality $89+110(h-0.25)\le 300\text{.}$ In context, this means that Jaylen would stay within their $\$300$ budget if there is only $2$ hours of labor.

Reading Questions 1.4.3 Reading Questions

1.

Is the equation in Example 1.4.3, $1.85+x=2.50\text{,}$ a linear equation?

2.

Give your own example of an equation in one variable that is not a linear equation.

3.

Do you believe it is possible for an inequality to have more than one solution? Do you believe it is possible for an equation to have more than one solution?

4.

There are two solutions to the equation in Example 1.4.7. What are they?

Exercises 1.4.4 Exercises

Review and Warmup

1.

Evaluate ${6-x}$ for $x = 0\text{.}$

2.

Evaluate ${-1-x}$ for $x = 2\text{.}$

3.

Evaluate ${-8x+5}$ for $x = 4\text{.}$

4.

Evaluate ${5x-8}$ for $x = 7\text{.}$

5.

Evaluate ${-5\!\left(t+9\right)}$ for $t = -2\text{.}$

6.

Evaluate ${-\left(x+6\right)}$ for $x = -9\text{.}$

7.

Evaluate the expression $\displaystyle \frac{1}{7} \big( x + 1 \big)^2 - 7$ when $x = -8\text{.}$

8.

Evaluate the expression $\displaystyle \frac{1}{3} \big( x + 2 \big)^2 - 4$ when $x = -5\text{.}$

9.

Evaluate the expression $-16t^{2}+64t+128$ when $t=-2\text{.}$

10.

Evaluate the expression $-16t^{2}+64t+128$ when $t=-4\text{.}$

Identifying Linear Equations and Inequalities

11.

Are the equations below linear equations in one variable?

$-4.12z=1$
- is
- is not
a linear equation in one variable.
$7+4y^{2}=24$
- is
- is not
a linear equation in one variable.
$\sqrt{1-0.5p}=9$
- is
- is not
a linear equation in one variable.
$x-8z^{2}=-11$
- is
- is not
a linear equation in one variable.
$4q+8=0$
- is
- is not
a linear equation in one variable.
$2\pi r=4\pi$
- is
- is not
a linear equation in one variable.

12.

Are the equations below linear equations in one variable?

$\sqrt{-3.3z-8}=1$
- is
- is not
a linear equation in one variable.
$1.55z=4$
- is
- is not
a linear equation in one variable.
$9z-2V^{2}=-26$
- is
- is not
a linear equation in one variable.
$7V^{2}-6=9$
- is
- is not
a linear equation in one variable.
$6q-16=-1$
- is
- is not
a linear equation in one variable.
$2\pi r=10\pi$
- is
- is not
a linear equation in one variable.

13.

Are the equations below linear equations in one variable?

$V\sqrt{30}=23$
- is
- is not
a linear equation in one variable.
$-0.44r=-7$
- is
- is not
a linear equation in one variable.
$q^{2}+z^{2}=34$
- is
- is not
a linear equation in one variable.
$\pi r^{2}=99\pi$
- is
- is not
a linear equation in one variable.
$4prV=-27$
- is
- is not
a linear equation in one variable.
$6-3p=-21$
- is
- is not
a linear equation in one variable.

14.

Are the equations below linear equations in one variable?

$z^{2}+y^{2}=-45$
- is
- is not
a linear equation in one variable.
$V\sqrt{30}=-64$
- is
- is not
a linear equation in one variable.
$9Vyz=-18$
- is
- is not
a linear equation in one variable.
$-2.43V=-52$
- is
- is not
a linear equation in one variable.
$15z-1=-34$
- is
- is not
a linear equation in one variable.
$\pi r^{2}=33\pi$
- is
- is not
a linear equation in one variable.

15.

Are the inequalities below linear inequalities in one variable?

$-4x^{2}-3z^{2}\gt1$
- is
- is not
a linear inequality in one variable.
$-2\geq5-10p$
- is
- is not
a linear inequality in one variable.
$6x^{2}-8V\gt-81$
- is
- is not
a linear inequality in one variable.

16.

Are the inequalities below linear inequalities in one variable?

$-3y^{2}-6q\leq44$
- is
- is not
a linear inequality in one variable.
$2\gt5-14x$
- is
- is not
a linear inequality in one variable.
$2p^{2}+6y^{2}\lt1$
- is
- is not
a linear inequality in one variable.

17.

Are the inequalities below linear inequalities in one variable?

$-3.9y\lt80$
- is
- is not
a linear inequality in one variable.
$\sqrt{4r}-14\lt5$
- is
- is not
a linear inequality in one variable.
$129\leq-5144y-2965q$
- is
- is not
a linear inequality in one variable.

18.

Are the inequalities below linear inequalities in one variable?

$-4.2z\geq-58$
- is
- is not
a linear inequality in one variable.
$-73\leq3916t+9643p$
- is
- is not
a linear inequality in one variable.
$\sqrt{4y}+2\leq4$
- is
- is not
a linear inequality in one variable.

Checking a Solution for an Equation

19.

Is $-1$ a solution for $x$ in the equation ${x-2} = {-1}\text{?}$

20.

Is $1$ a solution for $x$ in the equation ${x-7} = {-5}\text{?}$

21.

Is $7$ a solution for $r$ in the equation ${-6-r} = {-13}\text{?}$

22.

Is $3$ a solution for $t$ in the equation ${-8-t} = {-11}\text{?}$

23.

Is $-7$ a solution for $t$ in the equation ${-9t+7} = {70}\text{?}$

24.

Is $6$ a solution for $x$ in the equation ${x-6} = {0}\text{?}$

25.

Is $-2$ a solution for $x$ in the equation ${8x-5} = {-7x-20}\text{?}$

26.

Is $-8$ a solution for $y$ in the equation ${-4y+10} = {-7y-14}\text{?}$

27.

Is $7$ a solution for $y$ in the equation ${8\!\left(y+11\right)} = {19y}\text{?}$

28.

Is $-3$ a solution for $y$ in the equation ${3\!\left(y-8\right)} = {11y}\text{?}$

29.

Is $-10$ a solution for $r$ in the equation ${4\!\left(r-13\right)} = {11\!\left(r+1\right)}\text{?}$

30.

Is $-4$ a solution for $r$ in the equation ${14\!\left(r+1\right)} = {5\!\left(r+10\right)}\text{?}$

31.

Is ${{\frac{1}{3}}}$ a solution for $x$ in the equation ${6x-3} = -2\text{?}$

32.

Is ${{\frac{17}{9}}}$ a solution for $x$ in the equation ${9x-10} = 7\text{?}$

33.

Is ${{\frac{5}{2}}}$ a solution for $x$ in the equation ${-{\frac{2}{3}}x+1} = {0}\text{?}$

34.

Is ${-{\frac{5}{6}}}$ a solution for $x$ in the equation ${-{\frac{4}{3}}x - {\frac{2}{3}}} = {{\frac{4}{9}}}\text{?}$

35.

Is ${3}$ a solution for $x$ in the equation ${-{\frac{10}{3}}x-8} = {{\frac{9}{4}}x - {\frac{355}{36}}}\text{?}$

36.

Is ${-{\frac{2}{9}}}$ a solution for $y$ in the equation ${{\frac{10}{3}}y+{\frac{9}{4}}} = {-{\frac{1}{4}}y - {\frac{111}{8}}}\text{?}$