10.2: Solving and Graphing Inequalities, and Writing Answers in Interval Notation

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Page ID: 45212

Victoria Dominguez, Cristian Martinez, and Sanaa Saykali
Citrus College via ASCCC Open Educational Resources Initiative

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To solve and graph inequalities:

Solve the inequality using the Properties of Inequalities from the previous section.
Graph the solution set on a number line.
Write the solution set in interval notation.

Example 10.2.1

Solve the inequality, graph the solution set on a number line and show the solution set in interval notation:

$−1 ≤ 2x − 5 < 7$
$x^2 + 7x + 10 < 0$
$−6 < x − 2 < 4$

Solution

$\begin{array} &&−1 ≤ 2x − 5 < 7 &\text{Example problem} \\ &−1 + 5 ≤ 2x − 5 + 5 < 7 + 5 &\text{The goal is to isolate the variable \(x$, so start by adding $5$ to all three regions in the inequality.} \\ &4 ≤ 2x < 12 &\text{Simplify.} \\ &\dfrac{4}{2} ≤ 2x^2 < \dfrac{4}{2} &\text{Divide all by $2$ to isolate the variable $x$.} \\ &2 ≤ x < 6 &\text{Final answer written in inequality/solution set form.} \\ &[2, 6) &\text{Final answer written in interval notation (see section on Interval Notation for more details)} \end{array} \)

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$\begin{array} &&x^2 + 7x + 10 < 0 &\text{Example problem} \\ &(x + 5)(x + 2) < 0 &\text{Factor the polynomial.} \\ &(x + 5)(x + 2) < 0 &\text{The product must be less than \(0$, which means that if $(x + 5) > 0$, then $(x + 2) < 0$. Similarly, if $(x + 5) < 0$, then $(x + 2) > 0$.} \\ &(x + 5) > 0 ∪ (x + 2) < 0 &\text{Find the intersection of each of these inequalities.} \\ &x > −5 ∪ x < −2 &\text{Find the intersection of each of these inequalities.} \end{array}\)

$\begin{array} &&\;\;\;−5 < x < −2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{Final answer written in inequality/solution set form.} \\ &\;\;\;(−5, −2) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{Final answer written in interval notation (see section on Interval Notation for more details).} \end{array}$

$\begin{array}&&−6 < x − 2 ≤ 4 &\text{Example problem} \\ &−6 + 2 < x − 2 + 2 ≤ 4 + 2 &\text{The goal is to isolate the variable \(x$, so start by adding $2$ to all three regions in the inequality.} \\ &−4 < x ≤ 6 &\text{Final answer written in inequality/solution set form.} \\ &(−4, 6] &\text{Final answer written in interval notation (see section on Interval Notation for more details).} \end{array}\)

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Exercise 10.2.1

Solve the inequalities, graph the solution sets on a number line and show the solution sets in interval notation:

$0 ≤ x + 1 ≤ 4$
$0 < 2(x − 1) ≤ 4$
$6 < 2(x − 1) < 12$
$x^2 − 6x − 16 < 0$
$2x^2 − x − 15 > 0$

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Example 10.2.1

Exercise 10.2.1