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Precalculus for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version

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Section 1.1 Linear Equations and Inequalities (EQ1)

Subsection 1.1.1 Activities

Remark 1.1.1.

Activity 1.1.2.

Solve the linear equations.
(a)
\(3x-8=5x+2\)
  1. \(\displaystyle x=2\)
  2. \(\displaystyle x=5\)
  3. \(\displaystyle x=-5\)
  4. \(\displaystyle x=-2\)
Answer.
C
(b)
\(5(3x-4)=2x-(x+3)\)
  1. \(\displaystyle x=\dfrac{17}{14}\)
  2. \(\displaystyle x=\dfrac{14}{17}\)
  3. \(\displaystyle x=\dfrac{23}{14}\)
  4. \(\displaystyle x=\dfrac{14}{23}\)
Answer.
A

Activity 1.1.3.

Solve the linear equation.
\begin{equation*} \dfrac{2}{3}x-8=\dfrac{5x+1}{6} \end{equation*}
(a)
Which equation is equivalent to \(\dfrac{2}{3}x-8=\dfrac{5x+1}{6}\) but does not contain any fractions?
  1. \(\displaystyle 12x-48=15x+3\)
  2. \(\displaystyle 3x-24=10x+2\)
  3. \(\displaystyle 4x-8=5x+1\)
  4. \(\displaystyle 4x-48=5x+1\)
Answer.
D
(b)
Use the simplified equation from part (a) to solve \(\dfrac{2}{3}x-8=\dfrac{5x+1}{6}\text{.}\)
  1. \(\displaystyle x=-17\)
  2. \(\displaystyle x=-\dfrac{26}{7}\)
  3. \(\displaystyle x=-9\)
  4. \(\displaystyle x=-49\)
Answer.
D

Activity 1.1.4.

It is not always the case that a linear equation has exactly one solution. Consider the following linear equations which appear similar, but their solutions are very different.
(a)
Which of these equations has one unique solution?
  1. \(\displaystyle 4(x-2)=4x+6\)
  2. \(\displaystyle 4(x-1)=4x-4\)
  3. \(\displaystyle 4(x-1)=x+4\)
Answer.
C
(b)
Which of these equations has no solutions?
  1. \(\displaystyle 4(x-2)=4x+6\)
  2. \(\displaystyle 4(x-1)=4x-4\)
  3. \(\displaystyle 4(x-1)=x+4\)
Answer.
A
(c)
Which of these equations has many solutions?
  1. \(\displaystyle 4(x-2)=4x+6\)
  2. \(\displaystyle 4(x-1)=4x-4\)
  3. \(\displaystyle 4(x-1)=x+4\)
Answer.
B
(d)
What happens to the \(x\) variable when a linear equation has no solution or many solutions?
Answer.
The \(x\) variable cancels leaving only constants.

Definition 1.1.5.

conditional equation.identity equationinconsistent equation.

Activity 1.1.6.

An inequality is a relationship between two values that are not equal.
(a)
What is the solution to the linear equation \(3x-1=5\text{?}\)
Answer.
\(x=2\)
(b)
Which of these values is a solution of the inequality \(3x-1 \ge 5\text{?}\)
  1. \(\displaystyle x=0\)
  2. \(\displaystyle x=2\)
  3. \(\displaystyle x=4\)
  4. \(\displaystyle x=10\)
Answer.
B, C, and D
(c)
Express the solution of the inequality \(3x-1 \ge 5\) in interval notation.
  1. \(\displaystyle (-\infty, 2]\)
  2. \(\displaystyle (-\infty, 2)\)
  3. \(\displaystyle (2,\infty) \)
  4. \(\displaystyle [2,\infty)\)
Answer.
D
(d)
Draw the solution to the inequality on a number line.
Answer.
D

Activity 1.1.7.

Let’s consider what happens to the inequality when the variable has a negative coefficient.
(a)
Which of these values is a solution of the inequality \(-x\lt 8\text{?}\)
  1. \(\displaystyle x=-10\)
  2. \(\displaystyle x=-8\)
  3. \(\displaystyle x=4\)
  4. \(\displaystyle x=10\)
Answer.
C and D
(b)
Solve the linear inequality \(-x\lt 8\text{.}\) How does your solution compare to the values chosen in part (a)?
Answer.
\(x>-8\)
(c)
Expression the solution of the inequality \(-x\lt 8\) in interval notation.
  1. \(\displaystyle (-\infty,-8]\)
  2. \(\displaystyle (-\infty,-8)\)
  3. \(\displaystyle (-8, \infty)\)
  4. \(\displaystyle [-8,\infty)\)
Answer.
C
(d)
Draw the solution to the inequality on a number line.
Answer.
C

Remark 1.1.8.

Activity 1.1.9.

Solve the following inequalities. Express your solution in interval notation and graphically on a number line.
(a)
\(-3x-1 \le 5\)
Answer.
\(x \ge -2\text{,}\) \([-2,\infty]\)
(b)
\(3(x+4) \gt 2x-1\)
Answer.
\(x \gt -13\text{,}\) \((-13,\infty)\)
(c)
\(-\dfrac{1}{2}x \ge -\dfrac{3}{4}+\dfrac{5}{4}x\)
Answer.
\(x \le \dfrac{3}{7}\text{,}\) \(\left(-\infty, \dfrac{3}{7} \right]\)

Definition 1.1.10.

A compound inequality includes multiple inequalities in one statement.

Activity 1.1.11.

Consider the statement \(3 \le x \lt 8\text{.}\) This really means that \(3 \le x\) and \(x \lt 8\text{.}\)
(a)
Which of the following inequalities are equivalent to the compound inequality \(3 \le 2x-3 \lt 8 \text{?}\)
  1. \(\displaystyle 3 \le 2x-3\)
  2. \(\displaystyle 3 \ge 2x-3\)
  3. \(\displaystyle 2x-3 \lt 8\)
  4. \(\displaystyle 2x-3 \gt 8\)
Answer.
A and C
(b)
Solve the inequality \(3 \le 2x-3\text{.}\)
  1. \(\displaystyle x \le 0\)
  2. \(\displaystyle x \ge 0\)
  3. \(\displaystyle x \le 3\)
  4. \(\displaystyle x \ge 3\)
Answer.
B
(c)
Solve the inequality \(2x-3 \lt 8\text{.}\)
  1. \(\displaystyle x \gt \dfrac{11}{2}\)
  2. \(\displaystyle x \lt \dfrac{11}{2}\)
  3. \(\displaystyle x \gt \dfrac{5}{2}\)
  4. \(\displaystyle x \lt \dfrac{5}{2}\)
Answer.
B
(d)
Which compound inequality describes how the two solutions overlap?
  1. \(\displaystyle 0 \le x \lt \dfrac{11}{2}\)
  2. \(\displaystyle 0 \le x \lt \dfrac{5}{2}\)
  3. \(\displaystyle \dfrac{5}{2} \lt x \le 3\)
  4. \(\displaystyle 3 \le x \lt \dfrac{11}{2}\)
Answer.
A
(e)
Draw the solution to the inequality on a number line.
Answer.
A

Remark 1.1.12.

Activity 1.1.13.

Solve the following inequalities. Express your solution in interval notation and graphically on a number line.
(a)
\(8 \lt -3x-1 \le 11\)
Answer.
\(-4 \le x \lt -3\text{,}\) \([-4,-3)\)
(b)
\(-6 \le \dfrac{x-12}{4} \lt -2\)
Answer.
\(-12 \le x \lt 4\text{,}\) \([-12,4)\)

Exercises 1.1.2 Exercises

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