2.3 Intervals and Solving Inequalities - Introduction to Calculus

Inequality and Interval Notation

When solving an equation we may get a single answer, or a number of answers that satisfy the equation.

Consider $3x-5=1$ ; only one value satisfies this equation.

But if we consider $x^2-1=3$ , more than one value satisfies this equation.

Inequality notation, like $1\le x<3$ , uses the symbols $\le$ and $\ge$ to indicate inclusion of an endpoint, and $<$ and $>$ to indicate exclusion of an endpoint.

A second notation is interval notation, for example, $x\in[1,3)$ , where a square (or closed) bracket indicates inclusion of an endpoint, and a round (or open) bracket indicates exclusion of an endpoint.

The infinity symbol $\infty$ is always accompanied by round brackets.

Move the endpoints and change the brackets. Watch the circles and the inequality update together.

Left endpoint1Right endpoint3

Include left endpoint Include right endpoint

Interval:

\displaystyle [1,\,3)

\displaystyle 1 \le x < 3

Equal endpoints give a single point only when both ends are included; otherwise the interval is empty.

QUESTION

Which interval represents

\displaystyle 1\le x<3

Write each of the following in interval notation.

(a) $\displaystyle 2\le x\le7$

(b) $\displaystyle x<9$

Write each of the following using inequalities.

(a) $\displaystyle x\in[3,6)$

(b) $\displaystyle x\in(-2,4)$

Unions and Intersections

It is possible to have ranges of values that are disjoint. We use the union symbol $\cup$ to include all of the values in any of the disjoint ranges. For example, $[-1,4)\cup[7,10)$ means $-1\le x<4$ or $7\le x<10$

The intersection symbol $\cap$ allows only the values that are common between intervals. For example, $[-1,6)\cap(2,7)$ means $(2,6)$

Change the endpoints, then compare union (either interval) with intersection (both intervals).

A · left endpoint-1A · right endpoint4

Include left endpoint of A Include right endpoint of A

\displaystyle [-1,\,4)

B · left endpoint7B · right endpoint10

Include left endpoint of B Include right endpoint of B

\displaystyle [7,\,10)

Operation

A ∪ B:

\displaystyle [-1,\,4)\cup[7,\,10)

\displaystyle A\cup B=[-1,\,4)\cup[7,\,10)

If intervals meet at one point, check whether that point is included before deciding whether they join or intersect.

QUESTION

What is

\displaystyle [0,2]\cap(2,4)

Express each of the following in interval notation.

(a) $\displaystyle \displaystyle -3\le x<\dfrac12$ or $\displaystyle 4<x<7$

(b) $\displaystyle 1\le x<5$ or $\displaystyle 3\le x<7$

Express each of the following in interval notation.

(a) $\displaystyle -2<x\le6$ and $\displaystyle 0<x<7$

(b) $\displaystyle x\in[0,5]\cap[3,5]$

Solving Inequalities

When solving inequalities, there are a few rules that we must follow:

When it comes to addition, subtraction, multiplication, and division, what you do to one side of the inequality, you must do to the other.
If you multiply or divide by a negative quantity, you must flip the inequality.
If both sides are positive or both sides are negative, then you can take the reciprocal of both sides, but you must flip the inequality.

Division by zero is undefined. Multiplication by zero loses information, so it does not preserve an equivalent inequality.

Start with 2 < 5. Multiply both sides by the same factor.

Multiplication factor1

\displaystyle 2<5

Positive factor: keep the direction.

Try a factor of −1, then move through zero. Never divide by zero.

QUESTION

After dividing

\displaystyle -2x\le6

\displaystyle -2

, what do you obtain?

Find all values of $\displaystyle x$ that satisfy the following.

(a) $\displaystyle -6x+7\ge8x$

(b) $\displaystyle \displaystyle -\dfrac52<4-2x\le1$

(d) $\displaystyle 3x^2+2<-4$

(e) $\displaystyle \sqrt{x-1}>4$

(f) $\displaystyle \log_{\scriptscriptstyle 2}(3x)\le-3$

Inequality and Interval Notation

When solving an equation we may get a single answer, or a number of answers that satisfy the equation.

Consider $3x-5=1$ ; only one value satisfies this equation.

But if we consider $x^2-1=3$ , more than one value satisfies this equation.

Inequality notation, like $1\le x<3$ , uses the symbols $\le$ and $\ge$ to indicate inclusion of an endpoint, and $<$ and $>$ to indicate exclusion of an endpoint.

The infinity symbol $\infty$ is always accompanied by round brackets.

Move the endpoints and change the brackets. Watch the circles and the inequality update together.

Left endpoint1Right endpoint3

Include left endpoint Include right endpoint

Interval:

\displaystyle [1,\,3)

\displaystyle 1 \le x < 3

Equal endpoints give a single point only when both ends are included; otherwise the interval is empty.

QUESTION

Which interval represents

\displaystyle 1\le x<3

Write each of the following in interval notation.

(a) $\displaystyle 2\le x\le7$

(b) $\displaystyle x<9$

Write each of the following using inequalities.

(a) $\displaystyle x\in[3,6)$

(b) $\displaystyle x\in(-2,4)$

Unions and Intersections

The intersection symbol $\cap$ allows only the values that are common between intervals. For example, $[-1,6)\cap(2,7)$ means $(2,6)$

Change the endpoints, then compare union (either interval) with intersection (both intervals).

A · left endpoint-1A · right endpoint4

Include left endpoint of A Include right endpoint of A

\displaystyle [-1,\,4)

B · left endpoint7B · right endpoint10

Include left endpoint of B Include right endpoint of B

\displaystyle [7,\,10)

Operation

A ∪ B:

\displaystyle [-1,\,4)\cup[7,\,10)

\displaystyle A\cup B=[-1,\,4)\cup[7,\,10)

If intervals meet at one point, check whether that point is included before deciding whether they join or intersect.

QUESTION

What is

\displaystyle [0,2]\cap(2,4)

Express each of the following in interval notation.

(a) $\displaystyle \displaystyle -3\le x<\dfrac12$ or $\displaystyle 4<x<7$

(b) $\displaystyle 1\le x<5$ or $\displaystyle 3\le x<7$

Express each of the following in interval notation.

(a) $\displaystyle -2<x\le6$ and $\displaystyle 0<x<7$

(b) $\displaystyle x\in[0,5]\cap[3,5]$

Solving Inequalities

When solving inequalities, there are a few rules that we must follow:

When it comes to addition, subtraction, multiplication, and division, what you do to one side of the inequality, you must do to the other.
If you multiply or divide by a negative quantity, you must flip the inequality.
If both sides are positive or both sides are negative, then you can take the reciprocal of both sides, but you must flip the inequality.

Division by zero is undefined. Multiplication by zero loses information, so it does not preserve an equivalent inequality.

Start with 2 < 5. Multiply both sides by the same factor.

Multiplication factor1

\displaystyle 2<5

Positive factor: keep the direction.

Try a factor of −1, then move through zero. Never divide by zero.

QUESTION

After dividing

\displaystyle -2x\le6

\displaystyle -2

, what do you obtain?

Find all values of $\displaystyle x$ that satisfy the following.

(a) $\displaystyle -6x+7\ge8x$

(b) $\displaystyle \displaystyle -\dfrac52<4-2x\le1$

(d) $\displaystyle 3x^2+2<-4$

(e) $\displaystyle \sqrt{x-1}>4$

(f) $\displaystyle \log_{\scriptscriptstyle 2}(3x)\le-3$

Intervals and Solving Inequalities

Inequality and Interval Notation

Include or exclude an endpoint

KNOWLEDGE QUIZ

Write interval notation

Write inequalities

Unions and Intersections

Combine intervals on a number line

KNOWLEDGE QUIZ

Combine intervals with a union

Find common interval values

Solving Inequalities

When does the inequality reverse?

KNOWLEDGE QUIZ

Solve six inequalities

Intervals and Solving Inequalities

Inequality and Interval Notation

Include or exclude an endpoint

KNOWLEDGE QUIZ

Write interval notation

Write inequalities

Unions and Intersections

Combine intervals on a number line

KNOWLEDGE QUIZ

Combine intervals with a union

Find common interval values

Solving Inequalities

When does the inequality reverse?

KNOWLEDGE QUIZ

Solve six inequalities