Inequalities Part 1 Activity

Section 3.3 Inequalities Part 1 Activity

Subsection 3.3.1 Solving Inequalities

Activity 3.3.1.

Solve this inequality. Express your answer in a simple inequality or in words.

\begin{equation*} -5t + 13 \lt 26 \end{equation*}

Solution

We can subtract \(13\) from both sides. This preserves in inequality.

\begin{equation*} -5t \lt 13 \end{equation*}

We can divide by \(-5\) on both sides. This reverses the inequality

\begin{equation*} t \lt \frac{13}{-5} = - \frac{13}{5} \end{equation*}

This solve the inequality, which descries all number less than \(-\frac{13}{5}\text{.}\)

Activity 3.3.2.

Solve this inequality. Express your answer in a simple inequality or in words.

\begin{equation*} 9p + 4 \gt 13p - 10 \end{equation*}

Solution

We can add 10 to both sides. This preserves the inequality.

\begin{equation*} 9p + 14 \gt 13p \end{equation*}

We can subtract \(9p\) from both sides. This preserves the inequality.

\begin{equation*} 14 \gt 4p \end{equation*}

We can divide by 4 on both sides. This preserves the inequality.

\begin{equation*} \frac{14}{4} \gt p \end{equation*}

This solves the inequality, which describes all number less than \(\frac{14}{2}\text{.}\)

Activity 3.3.3.

Solve this inequality. Express your answer in a simple inequality or in words.

\begin{equation*} \sqrt{4t-5} \lt 100 \end{equation*}

Solution

We can square both sides. Since both sides are positive, this will preserve the inequality.

\begin{equation*} 4t-5 \lt 10000 \end{equation*}

Then we can add 5 to both sides. This preserves the inequality.

\begin{equation*} 4t \lt 10005 \end{equation*}

Then we can divide both sides by 4. This preserves the inequality.

\begin{equation*} t \lt \frac{10005}{4} \end{equation*}

This gives us a range of solutions: all numbers less than \(\frac{10005}{4}\text{.}\) However, we also have to make sure the square root is defined. This means we also have

\begin{equation*} 4t - 5 \geq 0 \end{equation*}

We can solve this similarly, by adding 5 then dividing by 4.

\begin{equation*} t \geq \frac{5}{4} \end{equation*}

This gives a lower bound for the variable \(t\text{.}\) All in all, we conclude that \(t\) must be greater than or equal to \(\frac{5}{4}\text{,}\) but less than \(\frac{10005}{4}\text{.}\)

Activity 3.3.4.

Solve this inequality. Express your answer in a simple inequality or in words.

\begin{equation*} t^2 + 1 \geq 5 \end{equation*}

Solution

We can subtract \(1\) from both sides, which preserves the inequality.

\begin{equation*} t^2 \geq 4 \end{equation*}

Then we can square root both sides. Since both sides are positive, this preserves the inequality.

\begin{equation*} t \geq 2 \end{equation*}

We also have to consider the possibility that \(t\) might have been negative. If so, its square would still be positive. In order for the square to exceed \(4\text{,}\) we need \(t \leq -2\text{.}\) Therefore, this inequality is solved by two groups of numbers: those greater than or equal to \(2\) and those less than or equal to \(-2\text{.}\)

Activity 3.3.5.

Solve this inequality. Express your answer in a simple inequality or in words.

\begin{equation*} y^2 + 4 \lt 2 \end{equation*}

Solution

We subtract \(4\) from both sides of the equation.

\begin{equation*} y^2 \lt -2 \end{equation*}

We could try to keep going here, but there is a problem. The right side is positive (all squares are positive), but it is supposed to be less than a negative number. This cannot happen. We conclude that there are no number which satisfy this inequality.

Subsection 3.3.2 Conceptual Questions

Activity 3.3.6.

What is an inequality?
When we work with equality, we can perform any operation as long as we apply it to both sides. Why isn't that true for inequalities? Why do some operations preserve, some reverse, and some to not respect inequalities at all?
What does it mean to solve an inequality?