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1.5E: Exercises

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    Practice Makes Perfect

    Graph Inequalities

    In the following exercises, graph each inequality on the number line and write in interval notation.

    1. \(x>3\)
    2. \(x\leq −0.5\)
    3. \(x\geq \frac{1}{3}\)
    4. \(x\leq 5\)
    5. \(x\geq −1.5\)
    6. \(x<−\frac{7}{3}\)
    7. \(−2<x<0\)
    8. \(−5\leq x<−3\)
    9. \(0\leq x\leq 3.5\)
    10. \(−4<x<2\)
    11. \(−5<x\leq −2\)
    12. \(−3.75\leq x\leq 0\)
    Answer
    1. The solution for x is greater than 3 on a number line has a left bracket 3 with shading to the right. The solution in interval notation is 3 to infinity within parentheses.
    2. The solution for x is less than or equal to negative 0.5 on a number line has a right bracket at negative 0.5 with shading to the left. The solution in interval notation is negative infinity to negative 0.5 within a parenthesis and a bracket.
    3. The solution for x is greater than or equal to one-third on a number line has a left bracket at one-third with shading to the right. The solution in interval notation is one-third to infinity within a bracket and a parenthesis.
    4. The solution for x is less than or equal to 5 on a number line has a right bracket with shading to the left. The solution in interval notation is negative infinity to 5 within a parenthesis and a bracket.
    5. The solution for x is greater than or equal to negative 1.5 on a number line has a left bracket with shading to the right. The solution in interval notation is negative 1.5 to infinity within a bracket and a parenthesis.
    6. The solution for x is less than negative seven-thirds on a number line has a right parenthesis with shading to the left. The solution in interval notation is negative infinity to negative seven-thirds within parentheses.
    7. Negative 2 is less than x which is less than 0. There is an open circle at negative 2 and an open circle at 0 and shading between negative 2 and 0 on the number line. The interval notation is negative 2 and 0 within parentheses.
    8. Negative 5 is less than or equal to x which is less than negative 3. There is a closed circle at negative 5 and an open circle at negative 3 and shading between negative 5 and negative 3 on the number line. The interval notation is negative 5 and negative 3 within a bracket and a parenthesis.
    9. 0 is less than or equal to x which is less than or equal to 3.5. There is a closed circle at 0 and a closed circle at 3.5 and shading between 0 and 3.5 on the number line. The interval notation is 0 and 3.5 within brackets.
    10. Negative 4 is less than x which is less than 2. There is a open circle at negative 4 and an open circle at 2 and shading between negative 4 and 2 on the number line. The interval notation is negative 4 and 2 within parentheses.
    11. Negative 5 is less than x which is less than or equal to 2. There is a open circle at negative 5 and a closed circle at negative 2 and shading between negative 5 and negative 2 on the number line. The interval notation is negative 5 and negative 2 within a parenthesis and a bracket.
    12. Negative 3.75 is less than or equal to x which is less than or equal to 0. There is a closed circle at negative 3.75 and a closed circle at 0 and shading between negative 3.75 and 0 on the number line. The interval notation is negative 3.75 and 0 within brackets.
     
    Solve Inequalities

    In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    1. \(b+\frac{7}{8}\geq \frac{1}{6}\)
    2. \(6y<48\)
    3. \(40<\frac{5}{8}k\)
    4. \(g−\frac{11}{12}<−\frac{5}{18}\)
    5. \(7s<−28\)
    6. \(\frac{9}{4}g\leq 36\)
    7. \(−8v\leq 96\)
    8. \(\frac{b}{−10}\geq 30\)
    9. \(−7d>105\)
    10. \(−18>\frac{q}{−6}\)
    Answer
    1. The solution is b is greater than or equal to negative seventeen twenty-fourths. The solution on a number line has a left bracket negative seventeen twenty-fourths with shading to the right. The solution in interval notation is negative seventeen twenty-fourths to infinity within a bracket and a parenthesis.
    2. The solution is y is less than 8. The solution on a number line has a right parenthesis at 8 with shading to the left. The solution in interval notation is negative infinity to 8 within parentheses.
    3. The solution is k is greater than 64. The solution on a number line has a left parenthesis at 64 with shading to the right. The solution in interval notation is 64 to infinity within parentheses.
    4. The solution is g is less than twenty-three thirty-sixths. The solution on a number line has a right parenthesis at twenty-three thirty-sixths with shading to the left. The solution in interval notation is negative infinity to twenty-three thirty-sixths within parentheses.
    5. The solution is s is less than negative 4. The solution on a number line has a right parenthesis at negative 4 with shading to the left. The solution in interval notation is negative infinity to negative 4 within parentheses.
    6. The solution is g is less than or equal to 16. The solution on a number line has a right bracket at 16 with shading to the left. The solution in interval notation is negative infinity to 16 within parenthesis and a bracket.
    7. The solution is v is greater than or equal to negative 12. The solution on a number line has a left bracket with shading to the right. The solution in interval notation is negative 12 to infinity within a bracket and a parenthesis.
    8. The solution is b is less than or equal to negative 300. The solution on a number line has a right bracket at negative 300 with shading to the left. The solution in interval notation is negative infinity to negative 300 within a parenthesis and a bracket.
    9. The solution is d is less than negative 15. The solution on a number line has a right parentheses with shading to the left. The solution in interval notation is negative infinity to negative 15 within parentheses.
    10. The solution is q is greater than 108. The solution on a number line has a left parentheses at 108 with shading to the right. The solution in interval notation is 108 to infinity within parentheses.
    Now with Interval Notation

    In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    1. \(5u\leq 8u−21\)
    2. \(9p>14p+18\)
    3. \(9y+5(y+3)<4y−35\)
    4. \(4k−(k−2)\geq 7k−26\)
    5. \(6n−12(3−n)\leq 9(n−4)+9n\)
    6. \(9u+5(2u−5)\geq 12(u−1)+7u\)
    7. \(12v+3(4v−1)\leq 19(v−2)+5v\)
    8. \(35k\geq −77\)
    9. \(18q−4(10−3q)<5(6q−8)\)
    10. \(−\frac{21}{8}y\leq −\frac{15}{28}\)
    Answer
    1. The solution is u is greater than or equal to 7. The solution on a number line has a left bracket at 7 with shading to the right. The solution in interval notation is 7 to infinity within a a bracket and a parenthesis.
    2. The solution is p is less than eighteen fifths. The solution on a number line has a right parenthesis at eighteen fifths with shading to the left. The solution in interval notation negative infinity to eighteen fifths within parentheses.
    3. The solution is y is less than negative 5. The solution on a number line has a right parenthesis at negative 5 with shading to the left. The solution in interval notation is negative infinity to negative 5 within parentheses.
    4. The solution is k is less than or equal to 7. The solution on a number line has a right bracket at 7with shading to the left. The solution in interval notation is negative infinity to 7 within a parenthesis and a bracket.
    5. The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.
    6. The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.
    7. The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.
    8. The solution is k is greater than or equal to negative eleven fifthss. The solution on a number line has a left bracket at negative eleven fifths with shading to the right. The solution in interval notation is negative eleven fifths to negative infinity within a bracket and a parenthesis.
    9. The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.
    10. The solution is y is greater than or equal to ten twenty-ninths. The solution on a number line has a left bracket at ten twenty-ninths with shading to the right. The solution in interval notation is ten twenty-ninths to infinity within a bracket and a parenthesis.

    This page titled 1.5E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Stanislav A. Trunov and Elizabeth J. Hale via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.