{"id":42,"date":"2019-04-29T13:58:33","date_gmt":"2019-04-29T13:58:33","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/mathtest\/chapter\/4-1-solve-and-graph-linear-inequalities\/"},"modified":"2020-10-06T16:14:20","modified_gmt":"2020-10-06T16:14:20","slug":"4-1-solve-and-graph-linear-inequalities","status":"web-only","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/mathtest\/chapter\/4-1-solve-and-graph-linear-inequalities\/","title":{"raw":"4.1 Solve and Graph Linear Inequalities","rendered":"4.1 Solve and Graph Linear Inequalities"},"content":{"raw":"https:\/\/www.youtube.com\/watch?v=Ky27IXLqI94\r\n\r\nWhen given an equation, such as [latex]x = 4[\/latex] or [latex]x = -5,[\/latex] there are specific values for the variable. However, with inequalities, there is a range of values for the variable rather than a defined value. To write the inequality, use the following notation and symbols:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Symbol<\/th>\r\nMeaning<\/th>\r\n<\/tr>\r\n
\"Right<\/td>\r\n> Greater than<\/td>\r\n<\/tr>\r\n
\"Right<\/td>\r\n\u2264\u00a0Greater than or equal to<\/td>\r\n<\/tr>\r\n
\"Left<\/td>\r\n< Less than<\/td>\r\n<\/tr>\r\n
\"Left<\/td>\r\n\u2265\u00a0Less than or equal to<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nhttps:\/\/www.youtube.com\/watch?v=P_-c9D6mjGA\r\n
\r\n

Example 4.1.1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nGiven a variable [latex]x[\/latex] such that [latex]x > 4[\/latex], this means that [latex]x[\/latex] can be as close to 4 as possible but always larger. For [latex]x > 4[\/latex], [latex]x[\/latex] can equal 5, 6, 7, 199. Even [latex]x =[\/latex] 4.000000000000001 is true, since [latex]x[\/latex] is larger than 4, so all of these are solutions to the inequality. The line graph of this inequality is shown below:\r\n\r\n\"x<\/span>\r\n\r\nWritten in interval notation, [latex]x > 4[\/latex] is shown as [latex](4, \\infty)[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 4.1.2<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nLikewise, if [latex]x < 3[\/latex], then [latex]x[\/latex] can be any value less than 3, such as 2, 1, \u2212102, even 2.99999999999. The line graph of this inequality is shown below:\r\n\r\n\"x<\/span>\r\n\r\nWritten in interval notation, [latex]x < 3[\/latex] is shown as [latex](-\\infty, 3)[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 4.1.3<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFor greater than or equal (\u2265) and less than or equal (\u2264), the inequality starts at a defined number and then grows larger or smaller. For [latex]x \\ge 4,[\/latex] [latex]x[\/latex] can equal 5, 6, 7, 199, or 4. The line graph of this inequality is shown below:\r\n\r\n\"x<\/span>\r\n\r\nWritten in interval notation, [latex]x \\ge 4[\/latex] is shown as [latex][4, \\infty)[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 4.1.4<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nIf [latex]x \\le 3[\/latex], then [latex]x[\/latex] can be any value less than or equal to 3, such as 2, 1, \u2212102, or 3. The line graph of this inequality is shown below:\r\n\r\n\"x<\/span>\r\n\r\nWritten in interval notation, [latex]x \\le 3[\/latex] is shown as [latex](-\\infty, 3].[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\nWhen solving inequalities, the direction of the inequality sign (called the sense) can flip over. The sense will flip under two conditions:\r\n\r\nFirst, the sense flips when the inequality is divided or multiplied by a negative. For instance, in reducing [latex]-3x <\u00a012[\/latex], it is necessary to divide both sides by \u22123. This leaves [latex]x > -4.[\/latex]\r\n\r\nSecond, the sense will flip over if the entire equation is flipped over. For instance, [latex]x\u00a0 >\u00a0 2[\/latex], when flipped over, would look like [latex]2 < x.[\/latex] In both cases, the 2 must be shown to be smaller than the [latex]x[\/latex], or the [latex]x[\/latex] is always greater than 2, no matter which side each term is on.\r\n
<\/div>\r\nFor questions 13 to 38, draw a graph for each inequality and give its interval notation.\r\n
    \r\n \t
  1. [latex]\\dfrac{x}{11}\\ge 10[\/latex]<\/li>\r\n \t
  2. [latex]-2 \\le \\dfrac{n}{13}[\/latex]<\/li>\r\n \t
  3. [latex]2 + r <\u00a0 3[\/latex]<\/li>\r\n \t
  4. [latex]\\dfrac{m}{5} \\le -\\dfrac{6}{5}[\/latex]<\/li>\r\n \t
  5. [latex]8+\\dfrac{n}{3}\\ge 6[\/latex]<\/li>\r\n \t
  6. [latex]11 > 8+\\dfrac{x}{2}[\/latex]<\/li>\r\n \t
  7. [latex]2 > \\dfrac{(a-2)}{5}[\/latex]<\/li>\r\n \t
  8. [latex]\\dfrac{(v-9)}{-4} \\le 2[\/latex]<\/li>\r\n \t
  9. [latex]-47 \\ge 8 -5x[\/latex]<\/li>\r\n \t
  10. [latex]\\dfrac{(6+x)}{12} \\le -1[\/latex]<\/li>\r\n \t
  11. [latex]-2(3+k) < -44[\/latex]<\/li>\r\n \t
  12. [latex]-7n-10 \\ge 60 [\/latex]<\/li>\r\n \t
  13. [latex]18 < -2(-8+p)[\/latex]<\/li>\r\n \t
  14. [latex]5 \\ge \\dfrac{x}{5} + 1[\/latex]<\/li>\r\n \t
  15. [latex]24\u00a0 \\ge -6(m - 6)[\/latex]<\/li>\r\n \t
  16. [latex]-8(n - 5) \\ge 0[\/latex]<\/li>\r\n \t
  17. [latex]-r -5(r - 6) < -18[\/latex]<\/li>\r\n \t
  18. [latex]-60\u00a0 \\ge -4( -6x - 3)[\/latex]<\/li>\r\n \t
  19. [latex]24 + 4b <\u00a0 4(1 + 6b)[\/latex]<\/li>\r\n \t
  20. [latex]-8(2 - 2n)\u00a0 \\ge -16 + n[\/latex]<\/li>\r\n \t
  21. [latex]-5v - 5 < -5(4v + 1)[\/latex]<\/li>\r\n \t
  22. [latex]-36 + 6x > -8(x + 2) + 4x[\/latex]<\/li>\r\n \t
  23. [latex]4 + 2(a + 5) < -2( -a - 4)[\/latex]<\/li>\r\n \t
  24. [latex]3(n + 3) + 7(8 - 8n) < 5n + 5 + 2[\/latex]<\/li>\r\n \t
  25. [latex]-(k - 2) > -k - 20[\/latex]<\/li>\r\n \t
  26. [latex]-(4 - 5p) + 3 \\ge -2(8 - 5p)[\/latex]<\/li>\r\n<\/ol>\r\nAnswer Key 4.1<\/a>","rendered":"