1.1 Integers and Rational Numbers

www.ck12.org Chapter 7. Literal Equations and InequalitiesExample 2: Check m ≤ 10 is a solution to 4m + 30 ≤ 70.Solution: If the solution set is true, any value less than or equal to 10 should make the original inequality true.Choose a value less than 10, say 4. Substitute this value for the variable m.4(4) + 3016 + 3046 ≤ 70The value found when m = 4 is less than 70. Therefore, the solution set is true.Why was the value 10 not chosen? Endpoints are not chosen when checking an inequality because the direction ofthe inequality needs to be tested. Special care needs to be taken when checking the solutions to an inequality.Solutions to one-variable inequalities can be graphed on a number line or in a coordinate plane.Example 3: Graph the solutions to t > 3 on a number line.Solution: The inequality is asking for all real numbers 3 or larger.You can also write inequalities given a number line of solutions.Example 4: Write the inequality pictured below.Solution: The value of four is colored in, meaning that four is a solution to the inequality. The red arrow indicatesvalues less than four. Therefore, the inequality is:x ≤ 4Inequalities that include the value are shown as ≤ or ≥. The line underneath the inequality stands for or equal to.We show this relationship by coloring in the circle above this value, as in the previous example. For inequalitieswithout the or equal to the circle above the value remains unfilled.Three Ways to Express Solutions to Inequalities1. Inequality notation: The answer is expressed as an algebraic inequality, such as d ≤ 1 2 .2. Interval notation: This notation uses brackets to denote the range of values in an inequality.• Square or closed brackets [ ] indicate that the number is included in the solution• Round or open brackets ( ) indicate that the number is not included in the solution.Interval notation also uses the concept of infinity ∞ and negative infinity −∞. For example, for all values of d thatare less than or equal to 1 2 , you could use set notation as follows: ( −∞, 1 }2119