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Math grade 9
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Quadratic inequalities</a>,<a href="https://www.purplemath.com/modules/ineqsolv3.htm">Other inequalities</a></em></p> <p>Solving linear inequalities is very similar to<a href="https://www.purplemath.com/modules/solvelin.htm">solving linear equations</a>, except for one small but important detail: you flip the inequality sign whenever you multiply or divide the inequality by a negative. The easiest way to show this is with some examples:</p> <ul>>td ><strong>2)</strong>>td ><strong>3)</strong>>td ><strong>4)</strong>>td ><strong>5)</strong><table><tbody><tr><td><strong>1)</XXXXXX></td><td><img XXX="https://www.XXXXXXXXXX.XXX/XXXXXXX/XXXXXXXX/ineqslva.XXX" alt="x + 3 < X, XXXX x < -1" width="123" XXXXXX="XX" style="width: XXXX; XXX-width: XXX%; XXXXXX: XXXX;"><p></p><p>XXXXXXXXXXX, the solution XX:</p><p><XXX XXX="https://XXX.XXXXXXXXXX.XXX/XXXXXXX/ineqsolv/ineqslvg.XXX" alt="graph XX XXXXXXXX" width="XXX" height="XX" style="width: auto; XXX-width: XXX%; height: XXXX;"></p></td><td <XXX="" XXXX="" XXXXXXXXXX="" between="" the="" XXXXXX="" XXXXXXXX="" "<i="">x+ X = 2" and this XXXXXX XXXXXXXXXXXX that I have a "less XXXX" sign, XXXXXXX XX an "equals" sign. XXX solution XXXXXX is XXXXXXX the same: XXXXXXXXXXXXX either side.<p>XXXX that XXX solution to a "XXXX than, but not XXXXX to" inequality is XXXXXXX XXXX a XXXXXXXXXXX (or XXXX an open dot) at the endpoint, indicating XXXX XXX endpoint is XXX XXXXXXXX within XXX solution.</p></td></tr><tr><td><img src="XXXXX://XXX.purplemath.XXX/XXXXXXX/XXXXXXXX/XXXXXXXX.gif" XXX="2 – x > X, then x < 2" width="115" height="XX" style="width: XXXX; XXX-width: 100%; XXXXXX: XXXX;"><p></p><p>XXXXXXXXXXX, the solution XX:</p><p><XXX XXX="https://www.XXXXXXXXXX.XXX/modules/XXXXXXXX/ineqslvh.XXX" alt="graph XX XXXXXXXX" width="172" XXXXXX="XX" style="width: auto; XXX-width: 100%; XXXXXX: auto;"></p></td><td <the="" only="" difference="" between="" the="" linear="" equation="" "X="" –<i="">x= 0" XXX this linear inequality XX the "XXXXXXX than" XXXX in XXXXX XX an "equals" XXXX.<p>XXXXXXXX "<em>x</em>" in the XXXXXXXX XXXX XXX "have" XX be XX XXX XXXX. However, it is often easier to picture what the XXXXXXXX means with XXX variable on XXX left. XXX't XX afraid to rearrange things to XXXX your taste.</p></td></tr><tr><td><img src="https://www.XXXXXXXXXX.XXX/XXXXXXX/XXXXXXXX/XXXXXXXX.gif" alt="XX + 6 >= XX – X, XXXX x <= –11" width="200" height="XXX" style="width: auto; max-width: XXX%; XXXXXX: XXXX;"><p></p><p>XXXXXXXXXXX, XXX XXXXXXXX is:</p><p><XXX src="https://XXX.XXXXXXXXXX.com/XXXXXXX/ineqsolv/XXXXXXXX.gif" alt="XXXXX XX XXXXXXXX" width="XXX" XXXXXX="XX" style="width: XXXX; XXX-width: XXX%; height: XXXX;"></p></td><td <XXX="" XXXX="" difference="" XXXXXXX="" the="" XXXXXX="" XXXXXXXX="" "4<i="">x+ X = X<em>x</em>– X" and XXXX inequality XX the "less than or XXXXX XX" XXXX in XXXXX XX a plain "XXXXXX" sign. The solution XXXXXX is exactly the same.<p>XXXX XXXX the solution XX a "less XXXX or equal XX" inequality XX XXXXXXX with a XXXXXX XXXXXXX (or else a closed XXX) at XXX XXXXXXXX, indicating XXXX the endpoint XX XXXXXXXX within XXX solution.</p></td></tr><tr><td><XXX XXX="https://XXX.XXXXXXXXXX.com/modules/XXXXXXXX/ineqslvd.gif" alt="2x > X, then x < X" width="XX" XXXXXX="35" style="width: XXXX; XXX-width: XXX%; height: XXXX;"><p></p><p>Graphically, the solution XX:</p><p><img src="https://XXX.purplemath.XXX/modules/XXXXXXXX/XXXXXXXX.gif" XXX="graph of XXXXXXXX" width="XXX" XXXXXX="32" style="width: auto; max-width: XXX%; XXXXXX: auto;"></p></td><td <XXX="" XXXXXXXX="" method="" here="" is="" to="" divide="" XXXX="" sides="" by="" a="" XXXXXXXX="" XXX.<p=""><p>XXXXXXXXX &XXXX; Elizabeth Stapel 1999-2011 XXX XXXXXX XXXXXXXX</p></td></tr><tr><td><img src="XXXXX://XXX.XXXXXXXXXX.com/modules/XXXXXXXX/XXXXXXXX.gif" alt="-2x > X, XXXX x < -2" width="127" height="59" style="width: XXXX; XXX-width: 100%; height: XXXX;"><p></p><p>Graphically, XXX XXXXXXXX is:</p><p><XXX src="XXXXX://www.purplemath.com/XXXXXXX/ineqsolv/XXXXXXXX.XXX" XXX="graphical solution" width="XXX" height="XX" style="width: XXXX; XXX-width: 100%; height: auto;"></p></td><td>XXXX is XXX special XXXX XXXXX XXXXX. XXXX I divided XX XXX<strong>negative</strong>two, I had to XXXX the inequality sign.</td></tr></tbody></table> </XX> <p>The rule for example5above XXXXX XXXXX unreasonable XX students XXX XXXXX XXXX XXXX see it. But think about inequalities XXXX XXXXXXX in XXXXX, XXXXXXX of XXXXXXXXX. XXX know XXXX XXX XXXXXX four XX larger XXXX XXX XXXXXX XXX:4 > X. XXXXXXXXXXX through this XXXXXXXXXX by–X, XX get–4 &XX; –2, XXXXX the XXXXXX XXXX shows is true:</p> <XX><p><XXX src="XXXXX://www.XXXXXXXXXX.XXX/modules/XXXXXXXX/ineqslvf.gif" XXX="number line" width="XXX" height="41" style="width: auto; max-width: 100%; height: auto;"></p></ul> <p>If XX hadn't flipped XXX XXXXXXXXXX, we XXXXX have XXXXX up XXXX "–X &XX; –2", XXXXX XXXXXXX isn't XXXX.</p>">
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