Review of Inequalities

1. Which of the $h$-values satisfy the following inequality?

$6 \geq \frac{h}{2}$

Choose all answers that apply:

A. $h=12$

B. $h = 14$

C. $h = 16$

2. Which of the $z$-values satisfy the following inequality?

$7 < \frac{z}{2}+3$

Choose all answers that apply:

A. $z = 8$

B. $z = 9$

C. $z = 10$

3. Which of the $f$-values satisfy the following inequality?

$f+8 \leq 12$

Choose all answers that apply:

A. $f=4$

B. $f=5$

C. $f=6$

4. Which of the $n$-values satisfy the following inequality?

$3 n-7 < 26$

Choose all answers that apply:

A. $n=10$

B. $n=11$

C. $n=12$

1. A. $h = 12$

Let's plug in $h=12$ and see if the inequality is true.

$\begin{aligned} 6 & \geq \frac{h}{2} \\ 6 & \stackrel {?}{\geq} \frac{12}{2} \\ 6 & \geq 6 \end{aligned}$

Yes, $h =12$ is a solution!

Now let's try $h =14$.

$\begin{aligned} &6 \geq \frac{h}{2} \\ &6 \stackrel {?}{\geq} \frac{14}{2} \\ &6 \not {\geq} 7 \end{aligned}$

No, $h=14$ is not a solution.

Let's try $h =16$.

$\begin{aligned} 6 & \geq \frac{h}{2} \\ 6 & \stackrel {?}{\geq} \frac{16}{2} \\ 6 & \not{\geq} 8 \end{aligned}$

No, $h = 16$ is not a solution.

The following $h$-value satisfies the inequality $6 \geq \frac{h}{2}$:

• $h = 12$

2. B.$z =9$, C. $z =10$

Let's plug in $z=8$ and see if the inequality is true.

$\begin{aligned} &7 < \frac{z}{2}+3 \\ &7  \stackrel {?}{ < } \frac{8}{2} + 3 \\ &7 \stackrel{?}{ < } 4+3 \\ &7 \nless 7 \\ \end{aligned}$

No, $z = 8$ is not a solution.

Now let's try $z=9$.

$\begin{aligned} &7 < \frac{z}{2}+3 \\ &7 \stackrel {?}{ < } \frac{9}{2} + 3 \\ &7 \stackrel{?}{ < } 4.5 +3 \\ &7 < 7.5 \\ \end{aligned}$

Yes, $z =9$ is a solution!

Let's try $z =10$.

$\begin{aligned} &7 < \frac{z}{2}+3 \\ &7 \stackrel {?}{ < } \frac{10}{2} + 3 \\ &7 \stackrel{?}{ < } 5 +3 \\ &7  < 7.5 \\ \end{aligned}$

Yes, $z =10$ is a solution!

The following $z$-values satisfy the inequality $7 < \frac{z}{2}+3$:

• $z =9$
• $z= 10$

3. A. $f=4$

Let's plug in $f=4$ and see if the inequality is true.

$\begin{aligned} f+8 & \leq 12 \\ 4+8 & \stackrel{?}{\leq} 12 \\ 12 & \leq 12 \end{aligned}$

Yes, $f=4$ is a solution.

Now let's try $f=5$.

$\begin{aligned} f+8 & \leq 12 \\ 5+8 & \stackrel {?} { \leq } 12 \\ 13 & \leq 12 \end{aligned}$

No, $f=5$ is not a solution.

Let's try $f=6$.

$\begin{aligned} f+8 & \leq 12 \\ 6+8 & \stackrel {?} { \leq } 12 \\ 14 & \leq 12 \end{aligned}$

No, $f=6$ is not a solution.

The following $f$-values satisfy the inequality $f+8 \leq 12:$

• $f=4$

4. A. $n=10$

Let's plug in $n=10$ and see if the inequality is true.

$\begin{aligned} 3 n-7 & < 26 \\ 3(10)-7 & \stackrel{?}{ < } 26 \\ 30-7 & \stackrel{?}{ < } 26 \\ 23 & < 26 \end{aligned}$

Yes, $n=10$ is a solution!

Now let's try $n=11$.

$\begin{aligned} 3 n-7 & < 26 \\ 3(11)-7 & \stackrel{?}{ < } 26 \\ 33-7 & \stackrel{?}{ < }26 \\ 23 & \nless 26 \end{aligned}$

No, $n=11$ is not a solution.

Let's try $n=12$.

$\begin{aligned} 3 n-7 & < 26 \\ 3(12)-7 & \stackrel{?}{ < } 26 \\ 36-7 & \stackrel{?}{ < } 26 \\ 29 & \nless 26 \end{aligned}$

No, $n=12$ is not a solution.

The following $n$-values satisfy the inequality $3 n-7 < 26$:

• $n=10$

1. Graph $x > 2$.

2. Graph $x \leq 5$.

3. Graph $x < 1$.

4. Graph $x \geq -1$.

1.

The $>$ is greater than symbol means "greater than and not including".

A filled circle means "including this number".

An open circle means "not including this number".

$x>2$ does not include $2$. We can draw an open circle at $2$.

Numbers to the right of $2$ are greater than $2$. We can draw an arrow to the right of the circle.

The graph of $x>2$ is:

2.

The $\leq$ symbol means "less than or equal to".

A filled circle means "including this number".

An open circle means "not including this number".

$x \leq 5$ includes $5$. We can draw a filled circle at $5$.

Numbers to the left of $5$ are less than $5$. We can draw an arrow to the left of the circle.

The graph of $x \leq 5$ is:

3.

The $<$ symbol means "less than and not including".

A filled circle means "including this number".

An open circle means "not including this number".

$x < 1$ does not include $1$. We can draw an open circle at $1$.

Numbers to the left of $1$ are less than $1$. We can draw an arrow to the left of the circle.

The graph of $x < 1$ is:

4.

The $\geq$ symbol means "greater than or equal to".

A filled circle means "including this number".

An open circle means "not including this number".

$x \geq -1$ includes $-1$. We can draw a filled circle at $-1$.

Numbers to the right of $-1$ are greater than $-1$. We can draw an arrow to the right of the circle.

The graph of $x \geq -1$ is:

1. The Russo-Japanese War was a conflict between Russia and Japan that started in the year $1904$.

Let $x$ represent any year. Write an inequality in terms of $x$ and $1904$ that is true only for values of $x$ that represent years before the start of the Russo-Japanese War.

2. John's goal is to have more than $-7$ dollars in his bank account by the end of the month.

The variable $d$ is the number of dollars in John's bank account at the end of the month.

Write an inequality in terms of $d$ that is true only if John meets his monthly goal.

3. Water boils at $212^{\circ}$ Fahrenheit.

Write an inequality that is true only for temperatures $(t)$ that are higher than the boiling point of water.

4.Two puppies, Ruth and Scoonsie, are playing together. Scoonsie weighs $11$ kilograms, and Ruth is lighter than Scoonsie.

Write an inequality that describes $r$, Ruth's weight in kilograms.

1. $x < 1904$ or $1904 > x$

Since the war started in $1904$, we want an inequality that represents years before $1904$

$x$ represents years, so we want this to be less than $1904$.

The inequality can be written $x < 1904$ or $1904 > x$.

2. $d > -7$

We want to represent account values with more than $-7$ dollars.

In mathematical terms, we want to represent account values with greater than $-7$ dollars.

$d > -7$

3. $t > 212$ or $212 < t$

Every temperature above $212^{\circ}$ is higher than the boiling point of water.

What relation does $t$ have to $212$ if $t$ is higher than the boiling point of water?

We want the inequality to say that $t$ is greater than $212$.

In other words, we want $t > 212$ or $212 < t$.

4. $r < 11$

Since Ruth is lighter than Scoonsie, she must be lighter than $11$ kilograms.

We want to write an inequality that shows weights less than $11$ kilograms.

$r < 11$