TBIL-Precal Linear Equations and Inequalities (EQ1)

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.

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(a)

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Which of these equations has one unique solution?

$4 (x - 2) = 4 x + 6$
$4 (x - 1) = 4 x - 4$
$4 (x - 1) = x + 4$

Answer.

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(b)

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Which of these equations has no solutions?

$4 (x - 2) = 4 x + 6$
$4 (x - 1) = 4 x - 4$
$4 (x - 1) = x + 4$

Answer.

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(c)

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Which of these equations has many solutions?

$4 (x - 2) = 4 x + 6$
$4 (x - 1) = 4 x - 4$
$4 (x - 1) = x + 4$

Answer.

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(d)

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What happens to the

x

variable when a linear equation has no solution or many solutions?

Answer.

The

x

variable cancels leaving only constants.

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Definition 1.1.5.

A linear equation with one unique solution is a conditional equation. A linear equation that is true for all values of the variable is an identity equation. A linear equation with no solutions is an inconsistent equation.

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Activity 1.1.6.

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An inequality is a relationship between two values that are not equal.

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(a)

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What is the solution to the linear equation

3 x - 1 = 5 ?

Answer.

x = 2

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(b)

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Which of these values is a solution of the inequality

3 x - 1 \geq 5 ?

$x = 0$
$x = 2$
$x = 4$
$x = 10$

Answer.

B, C, and D

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(c)

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Express the solution of the inequality

3 x - 1 \geq 5

in interval notation.

$(- \infty, 2]$
$(- \infty, 2)$
$(2, \infty)$
$[2, \infty)$

Answer.

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(d)

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Draw the solution to the inequality on a number line.

Answer.

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Activity 1.1.7.

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Let’s consider what happens to the inequality when the variable has a negative coefficient.

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(a)

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Which of these values is a solution of the inequality

- x < 8 ?

$x = - 10$
$x = - 8$
$x = 4$
$x = 10$

Answer.

C and D

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(b)

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Solve the linear inequality

- x < 8 .

How does your solution compare to the values chosen in part (a)?

Answer.

x > - 8

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(c)

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Expression the solution of the inequality

- x < 8

in interval notation.

$(- \infty, - 8]$
$(- \infty, - 8)$
$(- 8, \infty)$
$[- 8, \infty)$

Answer.

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(d)

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Draw the solution to the inequality on a number line.

Answer.

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Remark 1.1.8.

You can treat solving linear inequalities, just like solving an equation. The one exception is when you multiply or divide by a negative value, reverse the inequality symbol.

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Activity 1.1.9.

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Solve the following inequalities. Express your solution in interval notation and graphically on a number line.

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(a)

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- 3 x - 1 \leq 5

Answer.

x \geq - 2,

[- 2, \infty]

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(b)

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3 (x + 4) > 2 x - 1

Answer.

x > - 13,

(- 13, \infty)

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(c)

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- \frac{1}{2} x \geq - \frac{3}{4} + \frac{5}{4} x

Answer.

x \leq \frac{3}{7},

(- \infty, \frac{3}{7}]

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Definition 1.1.10.

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A compound inequality includes multiple inequalities in one statement.

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Activity 1.1.11.

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Consider the statement

3 \leq x < 8 .

This really means that

3 \leq x

and

x < 8 .

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(a)

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Which of the following inequalities are equivalent to the compound inequality

3 \leq 2 x - 3 < 8 ?

$3 \leq 2 x - 3$
$3 \geq 2 x - 3$
$2 x - 3 < 8$
$2 x - 3 > 8$

Answer.

A and C

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(b)

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Solve the inequality

3 \leq 2 x - 3 .

$x \leq 0$
$x \geq 0$
$x \leq 3$
$x \geq 3$

Answer.

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(c)

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Solve the inequality

2 x - 3 < 8 .

$x > \frac{11}{2}$
$x < \frac{11}{2}$
$x > \frac{5}{2}$
$x < \frac{5}{2}$

Answer.

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(d)

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Which compound inequality describes how the two solutions overlap?

$0 \leq x < \frac{11}{2}$
$0 \leq x < \frac{5}{2}$
$\frac{5}{2} < x \leq 3$
$3 \leq x < \frac{11}{2}$

Answer.

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(e)

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Draw the solution to the inequality on a number line.

Answer.

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Remark 1.1.12.

Solving a compound linear inequality, uses the same methods as a single linear inequality ensuring that you perform the same operations on all three parts. Alternatively, you can break the compound inquality up into two and solve separately.

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