2.8 - Linear inequalities and number lines

2.8 - Linear inequalities and number lines

Describe sets of possible values with inequalities, solve linear inequalities and show the answers on number lines. Learn how endpoint circles and negative division change what your answer means.

Inequality symbols and solution sets

A game allows values satisfying −2≤x<3-2\leq x<3. How many choices does that give: five, or infinitely many? The answer depends on whether the game allows only integers or any real number.

You already know

Integers are whole numbers, including zero and negative whole numbers. A real-number interval also contains fractions and decimals between its endpoints.

An equation usually asks for a value at which two expressions are equal. An inequality asks for values in an order relationship.

Inequality symbols

x<3x<3 means strictly less than 33; x>3x>3 means strictly greater than 33. The symbols ≤\leq and ≥\geq allow equality too. Thus 3≤33\leq3 is true, but 3<33<3 is false. “At most” means ≤\leq; “at least” means ≥\geq.

A double inequality joins two conditions with “and”. Read −2≤x<3-2\leq x<3 as “xx is at least −2-2, and xx is less than 33”. Its solution set contains every permitted value. These symbols and their meanings are not in either tier's formulae sheet: learn them.

Worked example 1: an integer restriction

Question. Write all the integers satisfying −2≤x<3-2\leq x<3.

Plan. Check each endpoint, then list the integers between them.

−2≤−2<3lower endpoint allowed3≮3upper endpoint excludedx=−2,−1,0,1,2list every integer inside\begin{aligned} -2\leq-2&<3 &&\text{lower endpoint allowed}\\ 3&\not<3 &&\text{upper endpoint excluded}\\ x&=-2,-1,0,1,2 &&\text{list every integer inside} \end{aligned}

Why stop at 22? The next integer is the excluded upper endpoint.

Answer. −2,−1,0,1,2-2,-1,0,1,2: five integers.

Check. Each is at least −2-2 and below 33; the next integers outside, −3-3 and 33, fail.

Now change only the upper sign so that equality is allowed there too.

Worked example 2: allowing the upper endpoint

Question. Write all the integers satisfying −2≤x≤3-2\leq x\leq3.

Plan. Keep the earlier list and check the newly included endpoint.

−2≤3≤3the upper endpoint now passesx=−2,−1,0,1,2,3include it in the list\begin{aligned} -2\leq3&\leq3 &&\text{the upper endpoint now passes}\\ x&=-2,-1,0,1,2,3 &&\text{include it in the list} \end{aligned}

Why does one line under a symbol matter? It changes whether the endpoint belongs to the set.

Answer. −2,−1,0,1,2,3-2,-1,0,1,2,3: six integers.

Check. −3-3 and 44 fail the lower and upper conditions respectively.

Without the word “integers”, a finite list would leave out permitted values such as 0.50.5.

Watch out

The wrong statement “−2≤x<3-2\leq x<3 means only −2,−1,0,1,2-2,-1,0,1,2” silently restricts xx to integers. 0.50.5 also satisfies the inequality. Give an integer list only when that restriction is stated.

Use both endpoint meanings in this check.

Open and closed number-line endpoints

A number line shows the whole real solution set, including values between its labelled ticks. Its endpoint circles tell you whether the bounds themselves belong.

Open and closed endpoints

An open circle excludes an endpoint, matching << or >>. A filled circle includes an endpoint, matching ≤\leq or ≥\geq. Join allowed values with a segment; use an arrow for a set continuing indefinitely.

An arrow at the edge of the page is not another endpoint. The two number lines show a bounded interval and an unbounded ray.

[DIAGRAM: asset_slug: m039_number_line_conventions; caption: An open lower and closed upper endpoint for minus 2 less than x at most 3, and a rightward ray for x at least 1; description: Two exact number lines, open at minus 2 and closed at 3 with the interval between, then closed at 1 with a rightward ray; file: m039-number-lines.png; recommended_method: matplotlib]
An open lower and closed upper endpoint for minus 2 less than x at most 3, and a rightward ray for x at least 1

Worked example 3: representing an interval

Question. Represent −2<x≤3-2<x\leq3 on a number line.

Plan. Test the two bounds, then mark the allowed interval.

−2≯−2open circle at −23≤3filled circle at 3−2<0≤3values between are allowed\begin{aligned} -2&\not>-2 &&\text{open circle at }-2\\ 3&\leq3 &&\text{filled circle at }3\\ -2&<0\leq3 &&\text{values between are allowed} \end{aligned}

Why join the circles? Both conditions must hold, so the solution is the segment between the bounds.

Answer. An open circle at −2-2, a filled circle at 33, and the segment between them, as in the top number line.

Check. −2-2 is excluded, 33 is included, and 0.50.5 lies on the allowed segment.

Change only the lower sign from strict to inclusive. The segment stays in the same place.

Worked example 4: changing one circle

Question. Represent −2≤x≤3-2\leq x\leq3 on a number line.

Plan. Check the new lower condition and keep the unchanged upper condition.

−2≤−2fill the lower circle3≤3keep the upper circle filled−2≤0≤3join the allowed values\begin{aligned} -2&\leq-2 &&\text{fill the lower circle}\\ 3&\leq3 &&\text{keep the upper circle filled}\\ -2&\leq0\leq3 &&\text{join the allowed values} \end{aligned}

Why fill both circles? Each bound now satisfies its equality condition.

Answer. Filled circles at −2-2 and 33, joined by a segment.

Check. Both endpoints and every value between them pass; −3-3 and 44 fail.

For x≥1x\geq1, there is no upper bound: fill the circle at 11 and draw a rightward ray, as in the lower diagram. For x<1x<1, use an open circle and a leftward ray. These conventions are required at both tiers and are not in either formulae sheet.

Watch out

The wrong instruction “fill the circle for x<1x<1” would include 11, yet 1<11<1 is false. Use an open circle. Circle style controls inclusion; ray direction controls which side is allowed.

On paper, make the representation before describing it in words.

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